This document discusses embedding and NP-complete problems related to equitable graph labelings. It begins by introducing concepts like cordial labeling, edge product cordial labeling, total edge product cordial labeling, difference cordial labeling, simply sequential labeling, total sequential cordial labeling, and divisor cordial labeling. It then presents proofs that any graph G can be embedded as an induced subgraph of graphs that admit these different labeling schemes, such as total edge product cordial graphs, difference cordial graphs, total sequential cordial graphs, and divisor cordial graphs. The document explores how to construct supergraphs of G that satisfy the conditions for these labeling types by adding vertices or edges to G in certain ways.
The document summarizes some new families of graphs that admit H-cordial and H2-cordial labelings. Specifically:
1) The graphs obtained by duplicating vertices or edges in the cycle Cn for n≥3 admit H2-cordial labelings.
2) The joint sum of two copies of the cycle Cn of even order produces an H2-cordial labeling graph.
3) The split graph of the cycle Cn for even n is H-cordial.
4) The shadow graph D2(Pn) of the path Pn is H-cordial for even n.
The document provides definitions, theorems with proofs, and examples
E-Cordial Labeling of Some Mirror GraphsWaqas Tariq
Let G be a bipartite graph with a partite sets V1 and V2 and G\' be the copy of G with corresponding partite sets V1\' and V2\' . The mirror graph M(G) of G is obtained from G and G\' by joining each vertex of V2 to its corresponding vertex in V2\' by an edge. Here we investigate E-cordial labeling of some mirror graphs. We prove that the mirror graphs of even cycle Cn, even path Pn and hypercube Qk are E-cordial graphs.
The document is a research paper that presents new results on odd harmonious graphs. It introduces the concepts of m-shadow graphs and m-splitting graphs. The paper proves that m-shadow graphs of paths and complete bipartite graphs are odd harmonious for all m ≥ 1. It also proves that n-splitting graphs of paths, stars and symmetric products of paths and null graphs are odd harmonious for all n ≥ 1. Additional families of graphs, including m-shadow graphs of stars and various graph constructions using paths, stars and their splitting graphs, are shown to admit odd harmonious labeling.
In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ 2, m ≡ 0 (mod 4) are odd harmonious graphs. Finally many new families of odd harmonious graphs are introduced.
A labeling of graph G is a mapping that carries a set of graph elements into a set of numbers (Usually positive integers) called labels. An edge magic labeling on a graph with p vertices and q edges will be defined as a one-to-one map taking the vertices and edges onto the integers 1,2,----,
p+q with the property that the sum of the label on an edge and the labels of its end vertices is constant independent of the choice of edge.
The document discusses characteristics of (γ, 3)-critical graphs. It begins by providing examples of (γ, 3)-critical graphs, such as the circulant graph C12 1, 4 and the Cartesian product Kt Kt . It then shows that a (γ, k)-critical graph is not necessarily (γ, k′)-critical for k ≠ k′ between 1 and 3. The document also verifies properties of (γ, 3)-critical graphs, such as not having vertices of degree 3. It concludes by proving characteristics about (γ, 3)-critical graphs that are paths, including that they have no vertices in V+ and satisfy other properties.
Cordial Labelings in the Context of TriplicationIRJET Journal
This document presents research on graph labelings for the extended triplicate graph of a ladder graph. It begins with an introduction to graph theory concepts like graph labelings and defines cordial, total cordial, product cordial, and total product cordial labelings. It then provides an algorithm to construct the extended triplicate graph of a ladder graph and proves that this graph admits cordial, total cordial, product cordial, and total product cordial labelings. Algorithms are presented for each type of labeling and proofs are given that the number of vertices and edges labeled 0 and 1 differ by at most 1, satisfying the conditions for these labeling types.
This document discusses theorems related to list coloring of graphs. It begins with definitions of graph coloring, list coloring, and other graph theory concepts. It then discusses several important theorems in the area, including:
- Thomassen's 5-list coloring theorem, which states that planar graphs are 5-list colorable.
- A result showing that the list chromatic number of a graph is bounded above by the maximum degree plus one.
- Examples showing that planar graphs require lists of size at least 4, as 4-list coloring is not always possible.
- A relationship between list coloring and sum-list coloring, and results relating their parameters.
The document summarizes some new families of graphs that admit H-cordial and H2-cordial labelings. Specifically:
1) The graphs obtained by duplicating vertices or edges in the cycle Cn for n≥3 admit H2-cordial labelings.
2) The joint sum of two copies of the cycle Cn of even order produces an H2-cordial labeling graph.
3) The split graph of the cycle Cn for even n is H-cordial.
4) The shadow graph D2(Pn) of the path Pn is H-cordial for even n.
The document provides definitions, theorems with proofs, and examples
E-Cordial Labeling of Some Mirror GraphsWaqas Tariq
Let G be a bipartite graph with a partite sets V1 and V2 and G\' be the copy of G with corresponding partite sets V1\' and V2\' . The mirror graph M(G) of G is obtained from G and G\' by joining each vertex of V2 to its corresponding vertex in V2\' by an edge. Here we investigate E-cordial labeling of some mirror graphs. We prove that the mirror graphs of even cycle Cn, even path Pn and hypercube Qk are E-cordial graphs.
The document is a research paper that presents new results on odd harmonious graphs. It introduces the concepts of m-shadow graphs and m-splitting graphs. The paper proves that m-shadow graphs of paths and complete bipartite graphs are odd harmonious for all m ≥ 1. It also proves that n-splitting graphs of paths, stars and symmetric products of paths and null graphs are odd harmonious for all n ≥ 1. Additional families of graphs, including m-shadow graphs of stars and various graph constructions using paths, stars and their splitting graphs, are shown to admit odd harmonious labeling.
In this paper, we show that the number of edges for any odd harmonious Eulerian graph is congruent to 0 or 2 (mod 4), and we found a counter example for the inverse of this statement is not true. We also proved that, the graphs which are constructed by two copies of even cycle Cn sharing a common edge are odd harmonious. In addition, we obtained an odd harmonious labeling for the graphs which are constructed by two copies of cycle Cn sharing a common vertex when n is congruent to 0 (mod 4). Moreover, we show that, the Cartesian product of cycle graph Cm and path Pn for each n ≥ 2, m ≡ 0 (mod 4) are odd harmonious graphs. Finally many new families of odd harmonious graphs are introduced.
A labeling of graph G is a mapping that carries a set of graph elements into a set of numbers (Usually positive integers) called labels. An edge magic labeling on a graph with p vertices and q edges will be defined as a one-to-one map taking the vertices and edges onto the integers 1,2,----,
p+q with the property that the sum of the label on an edge and the labels of its end vertices is constant independent of the choice of edge.
The document discusses characteristics of (γ, 3)-critical graphs. It begins by providing examples of (γ, 3)-critical graphs, such as the circulant graph C12 1, 4 and the Cartesian product Kt Kt . It then shows that a (γ, k)-critical graph is not necessarily (γ, k′)-critical for k ≠ k′ between 1 and 3. The document also verifies properties of (γ, 3)-critical graphs, such as not having vertices of degree 3. It concludes by proving characteristics about (γ, 3)-critical graphs that are paths, including that they have no vertices in V+ and satisfy other properties.
Cordial Labelings in the Context of TriplicationIRJET Journal
This document presents research on graph labelings for the extended triplicate graph of a ladder graph. It begins with an introduction to graph theory concepts like graph labelings and defines cordial, total cordial, product cordial, and total product cordial labelings. It then provides an algorithm to construct the extended triplicate graph of a ladder graph and proves that this graph admits cordial, total cordial, product cordial, and total product cordial labelings. Algorithms are presented for each type of labeling and proofs are given that the number of vertices and edges labeled 0 and 1 differ by at most 1, satisfying the conditions for these labeling types.
This document discusses theorems related to list coloring of graphs. It begins with definitions of graph coloring, list coloring, and other graph theory concepts. It then discusses several important theorems in the area, including:
- Thomassen's 5-list coloring theorem, which states that planar graphs are 5-list colorable.
- A result showing that the list chromatic number of a graph is bounded above by the maximum degree plus one.
- Examples showing that planar graphs require lists of size at least 4, as 4-list coloring is not always possible.
- A relationship between list coloring and sum-list coloring, and results relating their parameters.
Embedding and np-Complete Problems for 3-Equitable GraphsWaqas Tariq
We present here some important results in connection with 3-equitable graphs. We prove that any graph G can be embedded as an induced subgraph of a 3-equitable graph. We have also discussed some properties which are invariant under embedding. This work rules out any possibility of obtaining a forbidden subgraph characterization for 3-equitable graphs.
A common fixed point of integral type contraction in generalized metric spacessAlexander Decker
This document presents a common fixed point theorem for two self-mappings S and T on a G-metric space X that satisfies a contractive condition of integral type. It begins with definitions related to G-metric spaces and contractive conditions. It then states Theorem 1.1, which proves that if S and T satisfy the given integral type contractive condition, along with other listed conditions, then S and T have a unique point of coincidence in X. If S and T are also weakly compatible, then they have a unique common fixed point. The proof of Theorem 1.1 is then provided.
This document discusses different geometric structures and distances that can be used for clustering probability distributions that live on the probability simplex. It reviews four main geometries: Fisher-Rao Riemannian geometry based on the Fisher information metric, information geometry based on Kullback-Leibler divergence, total variation distance and l1-norm geometry, and Hilbert projective geometry based on the Hilbert metric. It compares how k-means clustering performs using distances derived from these different geometries on the probability simplex.
Clustering in Hilbert geometry for machine learningFrank Nielsen
- The document discusses different geometric approaches for clustering multinomial distributions, including total variation distance, Fisher-Rao distance, Kullback-Leibler divergence, and Hilbert cross-ratio metric.
- It benchmarks k-means clustering using these four geometries on the probability simplex, finding that Hilbert geometry clustering yields good performance with theoretical guarantees.
- The Hilbert cross-ratio metric defines a non-Riemannian Hilbert geometry on the simplex with polytopal balls, and satisfies information monotonicity properties desirable for clustering distributions.
The document introduces new classes of odd graceful graphs called m-shadow graphs and m-splitting graphs. It proves that m-shadow graphs of paths, complete bipartite graphs, and symmetric products of paths and null graphs are odd graceful. It also proves that m-splitting graphs of paths, stars, and symmetric products of paths and null graphs are odd graceful. Examples are provided to illustrate the theories.
Pullbacks and Pushouts in the Category of Graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
DISTANCE TWO LABELING FOR MULTI-STOREY GRAPHSgraphhoc
An L (2, 1)-labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k }such that |f(x)-f(y)| ≥2 if d(x, y) =1 and | f(x)- f(y)| ≥1 if d(x, y) =2. The L (2, 1)-labeling number λ (G) or span of G is the smallest k such that there is a f with
max {f (v) : vє V(G)}= k. In this paper we introduce a new type of graph called multi-storey graph. The distance two labeling of multi-storey of path, cycle, Star graph, Grid, Planar graph with maximal edges and its span value is determined. Further maximum upper bound span value for Multi-storey of simple
graph are discussed.
The document discusses various graph labeling concepts such as Z3-vertex magic total labeling, Z3-edge magic total labeling, and total magic cordial labeling. It proves that these labelings exist for the extended duplicate graph of a comb graph and the middle graph of the extended duplicate graph of a path graph. An algorithm is provided to obtain an n-edge magic labeling for the extended duplicate graph of a comb graph. The document also discusses the structures of these graphs.
Fixed point result in menger space with ea propertyAlexander Decker
This document presents a fixed point theorem for four self-maps in a Menger space. It begins by defining key concepts related to Menger spaces including probabilistic metric spaces, t-norms, neighborhoods, convergence, Cauchy sequences, and completeness. It then introduces properties like weakly compatible maps, property (EA), and JSR mappings. The main result, Theorem 3.1, proves the existence of a common fixed point for four self-maps under conditions that the map pairs satisfy a common property (EA) and are closed, JSR mappings satisfying an inequality involving the probabilistic distance functions.
Information geometry: Dualistic manifold structures and their usesFrank Nielsen
Information geometry: Dualistic manifold structures and their uses
by Frank Nielsen
Talk given at ICML GIMLI2018
http://gimli.cc/2018/
See tutorial at:
https://arxiv.org/abs/1808.08271
``An elementary introduction to information geometry''
An elementary introduction to information geometryFrank Nielsen
This document provides an elementary introduction to information geometry. It discusses how information geometry generalizes concepts from Riemannian geometry to study the geometry of decision making and model fitting. Specifically, it introduces:
1. Dually coupled connections (∇, ∇*) that are compatible with a metric tensor g and define dual parallel transport on a manifold.
2. The fundamental theorem of information geometry, which states that manifolds with dually coupled connections (∇, ∇*) have the same constant curvature.
3. Examples of statistical manifolds with dually flat geometry that arise from Bregman divergences and f-divergences, making them useful for modeling relationships between probability distributions
The document discusses Wythoff constructions and l1-embeddings. It begins by introducing Wythoff constructions, which generate new complexes from an original complex based on subsets of face dimensions. It then discusses l1-embeddings, which isometrically embed graphs into l1 metric spaces. Regular polytopes and tilings are provided as examples that embed into hypercubes or half-cubes through l1-embeddings. Embeddability can be tested using properties like being bipartite and satisfying hypermetric inequalities.
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
function f is said to be an even harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2q and the induced function f* from the edges ofG to {0, 2…….….2(q-1)} defined by f* (uv) = f (u) +f (v) (mod 2q) is bijective. The graph G is said to have an even harmonious labeling. In this paper the even harmonious labeling of a class of graph namely H (2n, 2t+1) is established.
The document describes a method for canonizing graphs of bounded treewidth in AC1 complexity. It presents the following:
1) Existing results showing canonization of bounded treewidth graphs is in P, TC1, TC2, and LogCFL complexity classes.
2) A new algorithm that canonizes bounded treewidth graphs in AC1 complexity by computing a tree decomposition of depth O(log n) and constructing a minimal description circuit of depth O(log n).
3) The algorithm works by computing descriptions for bags of the tree decomposition in parallel, sorting descriptions, and recursively combining descriptions while maintaining a circuit depth of O(log n).
This document contains 26 multiple choice questions and their answers related to graph theory. It begins by defining key graph theory terms like graphs, vertices, edges, simple graphs, and applications of graph theory. It then discusses incidence, adjacency, degrees, finite and infinite graphs, isolated and pendant vertices, null graphs, and multigraphs. The document also defines complete graphs, regular graphs, cycles, isomorphism, subgraphs, walks, paths, circuits, connectivity, components, Euler graphs, Hamiltonian circuits/paths, trees, properties of trees, distance in trees, eccentricity, center, distance metric, radius, diameter, rooted trees, and binary rooted trees.
This document summarizes research on edge coloring of complement fuzzy graphs. It defines fuzzy graphs and complement fuzzy graphs. It presents an algorithm to find the complement of any fuzzy graph in O(n2) time. It then defines a coloring function based on α-cut to color the edges of the complement fuzzy graph. The paper provides an example of a fuzzy graph G, calculates its complement Gc, and uses the coloring function to find the chromatic number of Gc in 3 or fewer colors.
Optimizing the parameters of wavelets for pattern matching using ga no restri...iaemedu
This document summarizes research optimizing the parameters of wavelets for pattern matching using genetic algorithms (GA). Wavelets have been used as tools for pattern matching, but traditionally parameter selection is done through trial and error. The paper proposes using GA to minimize the error between a pattern (in this case a sine wave) and a designed wavelet by optimizing the wavelet's parameters. It describes parametric wavelet design for filter lengths of 4, 6, 8, and 10 coefficients, and outlines the basic genetic algorithm used to optimize the parameters to best match the sine wave pattern. The results of applying this GA-based approach to parameter optimization are presented.
Fractal Reconfigurable Multiband Communicating Antenna for Cognitive RadioIOSR Journals
This document summarizes a research paper that proposes a novel fractal reconfigurable multiband antenna design for cognitive radio applications. The antenna structure combines a triangular fractal patch with parasitic elements and PIN diode switches. The switches can be activated or deactivated to change the electrical lengths and alter the current flow, dynamically changing the antenna's resonance frequencies. Simulation results show the antenna achieves frequency reconfigurability, operating at different resonant frequencies depending on the states of the switches. The proposed antenna design provides a compact, multiband solution that is well-suited for cognitive radio applications.
This document discusses performance of matching algorithms for signal approximation. It begins by introducing matching pursuit algorithms like Orthogonal Matching Pursuit (OMP) and Stagewise Orthogonal Matching Pursuit (StOMP) which are greedy algorithms that approximate sparse signals. It then describes the Non-Negative Least Squares algorithm which solves non-negative least squares problems. Finally, it discusses Extranious Equivalent Detection (EED), a modification of OED that incorporates non-negativity of representations by using a non-negative optimization technique instead of orthogonal projection.
Embedding and np-Complete Problems for 3-Equitable GraphsWaqas Tariq
We present here some important results in connection with 3-equitable graphs. We prove that any graph G can be embedded as an induced subgraph of a 3-equitable graph. We have also discussed some properties which are invariant under embedding. This work rules out any possibility of obtaining a forbidden subgraph characterization for 3-equitable graphs.
A common fixed point of integral type contraction in generalized metric spacessAlexander Decker
This document presents a common fixed point theorem for two self-mappings S and T on a G-metric space X that satisfies a contractive condition of integral type. It begins with definitions related to G-metric spaces and contractive conditions. It then states Theorem 1.1, which proves that if S and T satisfy the given integral type contractive condition, along with other listed conditions, then S and T have a unique point of coincidence in X. If S and T are also weakly compatible, then they have a unique common fixed point. The proof of Theorem 1.1 is then provided.
This document discusses different geometric structures and distances that can be used for clustering probability distributions that live on the probability simplex. It reviews four main geometries: Fisher-Rao Riemannian geometry based on the Fisher information metric, information geometry based on Kullback-Leibler divergence, total variation distance and l1-norm geometry, and Hilbert projective geometry based on the Hilbert metric. It compares how k-means clustering performs using distances derived from these different geometries on the probability simplex.
Clustering in Hilbert geometry for machine learningFrank Nielsen
- The document discusses different geometric approaches for clustering multinomial distributions, including total variation distance, Fisher-Rao distance, Kullback-Leibler divergence, and Hilbert cross-ratio metric.
- It benchmarks k-means clustering using these four geometries on the probability simplex, finding that Hilbert geometry clustering yields good performance with theoretical guarantees.
- The Hilbert cross-ratio metric defines a non-Riemannian Hilbert geometry on the simplex with polytopal balls, and satisfies information monotonicity properties desirable for clustering distributions.
The document introduces new classes of odd graceful graphs called m-shadow graphs and m-splitting graphs. It proves that m-shadow graphs of paths, complete bipartite graphs, and symmetric products of paths and null graphs are odd graceful. It also proves that m-splitting graphs of paths, stars, and symmetric products of paths and null graphs are odd graceful. Examples are provided to illustrate the theories.
Pullbacks and Pushouts in the Category of Graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
International Refereed Journal of Engineering and Science (IRJES)irjes
International Refereed Journal of Engineering and Science (IRJES) is a leading international journal for publication of new ideas, the state of the art research results and fundamental advances in all aspects of Engineering and Science. IRJES is a open access, peer reviewed international journal with a primary objective to provide the academic community and industry for the submission of half of original research and applications
DISTANCE TWO LABELING FOR MULTI-STOREY GRAPHSgraphhoc
An L (2, 1)-labeling of a graph G (also called distance two labeling) is a function f from the vertex set V (G) to the non negative integers {0,1,…, k }such that |f(x)-f(y)| ≥2 if d(x, y) =1 and | f(x)- f(y)| ≥1 if d(x, y) =2. The L (2, 1)-labeling number λ (G) or span of G is the smallest k such that there is a f with
max {f (v) : vє V(G)}= k. In this paper we introduce a new type of graph called multi-storey graph. The distance two labeling of multi-storey of path, cycle, Star graph, Grid, Planar graph with maximal edges and its span value is determined. Further maximum upper bound span value for Multi-storey of simple
graph are discussed.
The document discusses various graph labeling concepts such as Z3-vertex magic total labeling, Z3-edge magic total labeling, and total magic cordial labeling. It proves that these labelings exist for the extended duplicate graph of a comb graph and the middle graph of the extended duplicate graph of a path graph. An algorithm is provided to obtain an n-edge magic labeling for the extended duplicate graph of a comb graph. The document also discusses the structures of these graphs.
Fixed point result in menger space with ea propertyAlexander Decker
This document presents a fixed point theorem for four self-maps in a Menger space. It begins by defining key concepts related to Menger spaces including probabilistic metric spaces, t-norms, neighborhoods, convergence, Cauchy sequences, and completeness. It then introduces properties like weakly compatible maps, property (EA), and JSR mappings. The main result, Theorem 3.1, proves the existence of a common fixed point for four self-maps under conditions that the map pairs satisfy a common property (EA) and are closed, JSR mappings satisfying an inequality involving the probabilistic distance functions.
Information geometry: Dualistic manifold structures and their usesFrank Nielsen
Information geometry: Dualistic manifold structures and their uses
by Frank Nielsen
Talk given at ICML GIMLI2018
http://gimli.cc/2018/
See tutorial at:
https://arxiv.org/abs/1808.08271
``An elementary introduction to information geometry''
An elementary introduction to information geometryFrank Nielsen
This document provides an elementary introduction to information geometry. It discusses how information geometry generalizes concepts from Riemannian geometry to study the geometry of decision making and model fitting. Specifically, it introduces:
1. Dually coupled connections (∇, ∇*) that are compatible with a metric tensor g and define dual parallel transport on a manifold.
2. The fundamental theorem of information geometry, which states that manifolds with dually coupled connections (∇, ∇*) have the same constant curvature.
3. Examples of statistical manifolds with dually flat geometry that arise from Bregman divergences and f-divergences, making them useful for modeling relationships between probability distributions
The document discusses Wythoff constructions and l1-embeddings. It begins by introducing Wythoff constructions, which generate new complexes from an original complex based on subsets of face dimensions. It then discusses l1-embeddings, which isometrically embed graphs into l1 metric spaces. Regular polytopes and tilings are provided as examples that embed into hypercubes or half-cubes through l1-embeddings. Embeddability can be tested using properties like being bipartite and satisfying hypermetric inequalities.
We study QPT (quasi-polynomial tractability) in the worst case setting of linear tensor product problems defined over Hilbert spaces. We prove QPT for algorithms that use only function values under three assumptions'
1. the minimal errors for the univariate case decay polynomially fast to zero,
2. the largest singular value for the univariate case is simple,
3. the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point.
The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces.
Joint work with Erich Novak
function f is said to be an even harmonious labeling of a graph G with q edges if f is an injection from the vertices of G to the integers from 0 to 2q and the induced function f* from the edges ofG to {0, 2…….….2(q-1)} defined by f* (uv) = f (u) +f (v) (mod 2q) is bijective. The graph G is said to have an even harmonious labeling. In this paper the even harmonious labeling of a class of graph namely H (2n, 2t+1) is established.
The document describes a method for canonizing graphs of bounded treewidth in AC1 complexity. It presents the following:
1) Existing results showing canonization of bounded treewidth graphs is in P, TC1, TC2, and LogCFL complexity classes.
2) A new algorithm that canonizes bounded treewidth graphs in AC1 complexity by computing a tree decomposition of depth O(log n) and constructing a minimal description circuit of depth O(log n).
3) The algorithm works by computing descriptions for bags of the tree decomposition in parallel, sorting descriptions, and recursively combining descriptions while maintaining a circuit depth of O(log n).
This document contains 26 multiple choice questions and their answers related to graph theory. It begins by defining key graph theory terms like graphs, vertices, edges, simple graphs, and applications of graph theory. It then discusses incidence, adjacency, degrees, finite and infinite graphs, isolated and pendant vertices, null graphs, and multigraphs. The document also defines complete graphs, regular graphs, cycles, isomorphism, subgraphs, walks, paths, circuits, connectivity, components, Euler graphs, Hamiltonian circuits/paths, trees, properties of trees, distance in trees, eccentricity, center, distance metric, radius, diameter, rooted trees, and binary rooted trees.
This document summarizes research on edge coloring of complement fuzzy graphs. It defines fuzzy graphs and complement fuzzy graphs. It presents an algorithm to find the complement of any fuzzy graph in O(n2) time. It then defines a coloring function based on α-cut to color the edges of the complement fuzzy graph. The paper provides an example of a fuzzy graph G, calculates its complement Gc, and uses the coloring function to find the chromatic number of Gc in 3 or fewer colors.
Optimizing the parameters of wavelets for pattern matching using ga no restri...iaemedu
This document summarizes research optimizing the parameters of wavelets for pattern matching using genetic algorithms (GA). Wavelets have been used as tools for pattern matching, but traditionally parameter selection is done through trial and error. The paper proposes using GA to minimize the error between a pattern (in this case a sine wave) and a designed wavelet by optimizing the wavelet's parameters. It describes parametric wavelet design for filter lengths of 4, 6, 8, and 10 coefficients, and outlines the basic genetic algorithm used to optimize the parameters to best match the sine wave pattern. The results of applying this GA-based approach to parameter optimization are presented.
Fractal Reconfigurable Multiband Communicating Antenna for Cognitive RadioIOSR Journals
This document summarizes a research paper that proposes a novel fractal reconfigurable multiband antenna design for cognitive radio applications. The antenna structure combines a triangular fractal patch with parasitic elements and PIN diode switches. The switches can be activated or deactivated to change the electrical lengths and alter the current flow, dynamically changing the antenna's resonance frequencies. Simulation results show the antenna achieves frequency reconfigurability, operating at different resonant frequencies depending on the states of the switches. The proposed antenna design provides a compact, multiband solution that is well-suited for cognitive radio applications.
This document discusses performance of matching algorithms for signal approximation. It begins by introducing matching pursuit algorithms like Orthogonal Matching Pursuit (OMP) and Stagewise Orthogonal Matching Pursuit (StOMP) which are greedy algorithms that approximate sparse signals. It then describes the Non-Negative Least Squares algorithm which solves non-negative least squares problems. Finally, it discusses Extranious Equivalent Detection (EED), a modification of OED that incorporates non-negativity of representations by using a non-negative optimization technique instead of orthogonal projection.
All optical single module phase encoded half-adder and halfsubtractor exploit...IOSR Journals
The development of all optical super fast arithmetic logic unit (ALU) is one of the most important
requirement for construction of super fast all optical processor. All optical half-adder and half-subtractor have
performed most of the essential functions of the arithmetic logic unit. Here in this paper the author has
proposed a scheme of single module for phase encoded simultaneous operation of half-addition and halfsubtraction
for the same input bits using four wave mixing (FWM) in semiconductor optical amplifier (SOA).
This type of single module for simultaneous operations using phase encoding based on the difference of phase
with the reference signal one through FWM in SOA will gives fast response to accomplish the demand of super
fast all optical processing
Determination of Propagation Path Loss and Contour Map for FUTA FM Radio Fede...IOSR Journals
FM signal propagation through the troposphere interacts with the terrain as obstacles and reflection
planes. To understand the degree of interaction, signal strength measurements of the 93.1MHz frequency
modulated Radio located at Federal University of Technology; Akure, Nigeria was carried out in the area
surrounding the station. The paper reviews the various models for predicting transmission loss and employed
the long rice irregular terrain model for its versatility for the study. The losses along the paths were determined
and this was compared with the path loss predicted by the irregular terrain model and this was highly
correlated. The result offers useful data for developing the contour map of the propagation loss which was
developed for the station. It was concluded that with the irregular terrain model predictions can be used for
accurate spectrum management in Nigeria
Prevalence of Hepatitis B Surface Antigen among Undergraduate Students of Gom...IOSR Journals
Incidence of Hepatitis B virus among healthy asymptomatic students in Gombe State University was determined, this was in an effort of providing baseline data on the diseases burden, and the possible risk factors associated with the infection in the study population. A total of 100 serum samples were collected from volunteers and screened using rapid immune chromatographic test kits for Hepatitis B surface antigen (HBsAg). The study revealed that 14% were HBsAg positive. The highest incidence rate of 18.2% (12) was recorded among the age group of 16-25 years, and males recorded the highest incidence rate of 20% (12), indicating that gender but not age might have greater influence on the infection (P= 0.05).
Review and Performance Comparison of Distributed Wireless Reprogramming Proto...IOSR Journals
Abstract:A Reprogramming service should be efficient, reliable and secured in Wireless sensor network.
Wireless reprogramming for wireless sensor network emphasize over the process of changing or improving the
functionality of simulation or existing code. For challenging and on demand security purpose, secure and
distributed routing protocols such as SDRP and ISDRP were developed. This paper reviews and compares the
propagation delay for two reprogramming protocols, SDRP and ISDRP, which based on hierarchy of energies
in network. Both are based on identity-based cryptography. But in the improved protocol the keys are
distributed to the network as per the sorting and communication capabilities to improve the broadcast or
communication nature of the network. Moreover, ISDRP demonstrates the security concepts, which deals over
the key encryption properties using heap sort algorithm and the confidentiality parameter is enhanced by
changing the private key values after certain interval of time for cluster head in respect to different public keys.
The ISDRP shows high efficiency rate clearly with the throughput and propagation results by implementation in
practice over SRDP.
Keywords: identity-based cryptography,ISDRP, heapsort algorithm, Reprogramming, SDRP, Wireless sensor
network.
Capacity Spectrum Method for RC Building with Cracked and Uncracked SectionIOSR Journals
one of the most widespread procedures for the assessment of building behavior, due to earthquake, is the Capacity Spectrum Method (CSM). In the scope of this procedure, capacity of the structure compares with the demands of earthquake ground motion on the structure. The capacity of the structure is represented by a nonlinear force-displacement curve, referred to as a pushover curve. The base shear forces and roof displacements are converted to equivalent spectral accelerations and spectral displacements, respectively, by means of coefficients that represent effective modal masses and modal participation factors. These spectral values define the capacity spectrum. The demands of the earthquake ground motion are represented by response spectra. A graphical construction that includes both capacity and demand spectra, results in an intersection of the two curves that estimates the performance of the structure to the earthquake. In this study, for determination of the performance levels, G+10 R.C.C. Building with cracked and uncracked section were taken. The structural Capacity of cracked and uncracked section compared with performance point value, which shows the structural capacity of building having cracked section is lesser than the uncracked section. Different modeling issues were analyzed to study the effect on Capacity of the structure with cracked and uncracked section for different position of Shear wall.
Soil-transmitted helminth infections in relation to the knowledge and practic...IOSR Journals
The relationship between soil-transmitted helminth infections and the knowledge and practice of preventive measures among school children in rural communities in Igbo-Eze South Local Government Area of Enugu State, South-Eastern Nigeria, was investigated. Stool samples were obtained from 1,296 school children (ages 4 – 15 years) from six schools randomly selected from the study area. Structured epidemiological questionnaires were administered to the children. Out of 1,296 school children examined, 106 (8.1 %) of the children were infected by soil-transmitted infections thus: 64 (4.9 %) with Ascarislumbricoides, 33 (2.5 %) with hookworm, and 9 (0.7 %) with Trichuristrichiura. There were significant differences in the prevalence of these infections (P < 0.05). Soil-transmitted helminth infections showed statistically significant (P < 0.05) relationships with knowledge and practice of preventive measures among school children in the study area. The study revealed that soil-transmitted helminth infections were abundant among school children of the study area, indicating the necessity of implementing control measures such as chemotherapy, provision of adequate sanitary facilities and safe drinking water.
To Study the Efficacy of Electromyographic Biofeedback Training on Dynamic Eq...IOSR Journals
Abstract: Cerebral palsy (CP) is caused by static lesion to a developing nervous system that primarily affects
motor function. Spastic motor involvement is characteristic of most of these individual.Dynamicequinus is a
common deformity that worsens the ambulatory ability of both diplegic and hemiplegic conditions. The use of
electromyographic (EMG) biofeedback has been suggested as a training tool to improve the ability to increase
activation of weak and partially paralyzed muscles and to decrease the activation of muscles affected by spasm
or spasticity without regard to specific diagnosis. However, very few studies have reported the effects of EMG
biofeedback on ankle function among children with spastic cerebral palsy .Objectives of the study was to
increase the activation of tibialis anterior and to improve the functional ambulation.40 subjects were made part
of the study on the basis of inclusion and exclusion criteria divided into two groups group A and B.Group A
received traditional physical therapy exercises and electromypgraphic biofeedback and group B received only
exercise program.The treatment duration was for 4weeks 3 sessions a week.The results were analysed using
statistical tests that were paired and unpaired t-test and mannwhitneytest.The results showed significant
improvement in the pre and post treatment.The conclusion of the study lended a favourable outlook to use
biofeedback training in treatment of CP children, to improve functional ambulation and gait. Keywords: Cerebral palsy, dynamic equinus deformity, gait, electromyography, biofeedback.
Effect of Regular Exercise on Prolactin Secretion: A Pilot StudyIOSR Journals
Abstract:
Introduction: Evidence suggested that exercise may affects release of prolactin hormone. Participation in
exercise may increase secretion of prolactin hormone and may give sharp decrease in secretion of prolactin and
not only that published work demanded that there is no effect of exercise on secretion of prolactin. In this
context the researcher intend to know whether exercise affects positively or negatively or not on the release of
prolactin hormone. Aim: Determine whether participation in exercise may increase the level of secretion of
prolactine hormone or not. Method: Only two female students 29 years aged were participated in this study.
They were regularly practiced yogic asana and pranayam for 1 hour per day in the evening, 6 days per week,
for 8 weeks. The level of prolactin hormone was assessed by CLI method. In the present study all the
measurements were done at the baseline and 8 weeks of exercise training. Simple percentage calculated from
the mean value to see the quantitative changes in secretion of prolactin due to participation in the exercise
training. Result: Pre test mean was 5.80 and post test mean was 17.63 which imply that 203.96% increase in
secretion of prolactin significantly. Discussion and Conclusion: Level of secretion increased may be due to
multiple neural pathways that influence PRL secretion converges on the hypothalamus from other parts of the
brain; the effect of exercise on the secretion of PRL may also reflect the action of different neural inputs on the
activity of the hypothalamic–pituitary axis.
Key word: Exercise; Prolactin.
Old wine in new wineskins: Revisiting counselling in traditional Ndebele and ...IOSR Journals
The document summarizes counselling in traditional Ndebele and Shona societies in Zimbabwe before the advent of western formal counselling. It discusses how counselling was an integral part of communities with various roles providing advice, such as close family friends, aunts, uncles, grandparents, traditional healers, and elders. Folklore and proverbs were also important means of imparting guidance. Counselling emphasized prevention and was provided informally and freely within a holistic community approach, rather than a professionalized, crisis-focused model as seen today. The document argues counselling is not new but an old practice now in a modernized form, representing "old wine in new wineskins."
Modeling & Testing Of Hybrid Composite LaminateIOSR Journals
This document summarizes research conducted on modeling and testing a hybrid composite laminate made of glass wool and epoxy resin. Four different types of laminates were prepared by varying the fiber orientation between layers. The laminates were tested for mechanical properties according to ASTM standards. Results showed that increasing cure time improved bonding strength. Changing fiber orientation between layers decreased strength due to delamination. The hybrid laminates were found to have higher strength to weight ratios than steel, indicating their usefulness in applications requiring both strength and light weight. Finite element analysis using ANSYS was also conducted to model stresses in the laminate.
Prevalence of anemia and socio-demographic factors associated with anemia amo...IOSR Journals
Objective: To study the prevalence of anemia and socio-demographic factors associated with anemia among pregnant women attending antenatal hospital. Material and methods: Total 100 pregnant women were selected for the study and the study was carried out from Sep 2011 to Jan 2012. Pretested and prestructured questionnaire was used to collect general information. Blood samples were collected by qualified technician for hemoglobin estimation. Socio-economic classification by B.G. Prasad was adopted. Classification of anemia by WHO was used. Chi-square test was used for statistical analysis. Results: Overall prevalence of anemia among the pregnant women was found to be 63%. Factors such as level of education and socio-economic status were found to be significantly associated with prevalence of anemia. Conclusion: There is a need for realization that health system should focus on various factors that contribute to the occurrence of anemia and include them as important mediators in the National Health Policy.
Effect of Planar Solid Shear Wall - Frame Arrangement on the Deformation Beha...IOSR Journals
This document summarizes a study that analyzed the deformation behavior of different planar solid shear wall-frame arrangements in a 15-story reinforced concrete office building subjected to wind loads. Four different arrangements of equivalent shear wall stiffnesses were considered. A 3D finite element analysis was performed using STAAD Pro software. The results showed that the arrangement with shear walls positioned parallel to and away from the wind direction centroidal axis of the frame had the minimum storey displacement, indicating it was the best arrangement for minimizing deformation. Internal forces like axial forces, moments and shears were also compared between the arrangements. There was a strong positive correlation between beam moment and maximum displacement.
The electronic band parameters calculated by the Triangular potential model f...IOSR Journals
This work reports on theoretical investigation of superlattices based on Cd1-xZnxS quantum dots
embedded in an insulating material. This system, assumed to a series of flattened cylindrical quantum dots with
a finite barrier at the boundary, is studied using the triangular potential. The electronic states and the effective
mass of 1 Γ miniband have been computed as a function of inter-quantum dot separation for different zinc
compositions. Calculations have been made for electrons, heavy holes and light holes. Results are discussed and
compared with those of the Kronig-Penney and sinusoidal potentials
A Modified H-shaped Microstrip Patch Antenna with Periodic Ground StructureIOSR Journals
The document proposes a modified H-shaped microstrip patch antenna with a periodic ground structure to achieve broadband capabilities. Simulation results show the proposed antenna design offers a bandwidth of 52.43% ranging from 3.165-5.565 GHz. The antenna also exhibits stable radiation patterns and a gain of over 2 dB across the entire operating bandwidth, making it suitable for applications such as WLAN, WiMAX, and Bluetooth. Parameter sweeps were performed to optimize the antenna geometry and achieve the desired impedance bandwidth and performance.
This document presents a scalable method for image classification using sparse coding and dictionary learning. It proposes parallelizing the computation of image similarity for faster recognition. Specifically, it distributes the task of measuring similarity between images among multiple cores in a cluster. Experimental results on a face recognition dataset show nearly linear speedup when balancing the dataset size and number of nodes. Reconstruction errors are used as a similarity measure, with dictionaries learned using K-SVD for each image. The proposed parallel method distributes this similarity computation process to achieve faster image classification.
This document discusses the development of a web-based decision support system (DSS) for monitoring and predicting water quality parameters important for outdoor microalgae cultivation. The DSS allows users to monitor temperature and salinity in real-time and predicts future values of these parameters based on historical data using a K-nearest neighbors algorithm. Validation tests found the DSS could accurately forecast temperature and salinity 96.98% and 98.92% of the time respectively. The DSS notifies users when parameter levels rise or fall outside standard ranges and provides suggestions to maintain optimal water quality.
This document summarizes an article about India's energy policy and the need to promote renewable energy sources. It discusses how India has vast renewable energy resources and the government has implemented various policies and incentives to promote greater renewable energy deployment. The key challenges are India's limited fossil fuel reserves, high fuel transportation costs, aging conventional power plants, need to rationalize power tariffs, and reduce transmission and distribution losses in the power sector. The government is aiming to source 10% of additional grid power capacity from renewable sources by 2012 to help address these issues through its renewable energy policies.
1) A single-phase matrix converter topology is presented that can synthesize a lower or higher DC output voltage from a given AC supply voltage. It uses an active pulse width modulation technique to maintain a continuous, sinusoidal input current that is in phase with the supply voltage, improving the input power factor.
2) Conventional rectifiers draw discontinuous current with high harmonics, resulting in poor power quality. The proposed single-phase matrix converter acting as a rectifier with active PWM can suppress harmonic current drawn by the rectifier load.
3) Simulation results show that for boost and buck rectification using the proposed single-phase matrix converter with active PWM technique, the supply current is sinusoidal and in phase
This document provides an introduction to graph theory, including basic terminology and concepts. It defines what a graph is mathematically as a collection of vertices and edges. It describes different types of graphs such as simple graphs, multiple graphs, weighted graphs, finite and infinite graphs, labeled graphs, and directed graphs. It also defines graph theory terms like adjacency, incidence, degree of a vertex, isomorphism, subgraphs, and graph operations like union and intersection. The document provides examples to illustrate each term and concept.
This document provides an introduction to graph theory, including basic terminology and concepts. It defines what a graph is mathematically as a collection of vertices and edges. It describes different types of graphs such as simple graphs, multiple graphs, weighted graphs, finite and infinite graphs, labeled graphs, and directed graphs. It also defines graph theory terms like adjacency, incidence, degree of a vertex, isomorphism, subgraphs, and graph operations like union and intersection. The document provides examples to illustrate each term and concept.
1. Graph and Graph Terminologiesimp.pptxswapnilbs2728
There are five main categories of graphs: simple graphs, multigraphs, pseudographs, directed graphs, and directed multigraphs. An undirected graph G consists of a set of vertices V and a set of edges E that connect the vertices. A directed graph consists of vertices V and directed edges E that have an initial and terminal vertex. There are several special types of simple graphs including complete graphs, cycles, wheels, and bipartite graphs.
FURTHER RESULTS ON ODD HARMONIOUS GRAPHSFransiskeran
In [1] Abdel-Aal has introduced the notions of m-shadow graphs and n-splitting graphs, for all m, n ≥ 1.
In this paper, we prove that, the m-shadow graphs for paths and complete bipartite graphs are odd
harmonious graphs for all m ≥ 1. Also, we prove the n-splitting graphs for paths, stars and symmetric
product between paths and null graphs are odd harmonious graphs for all n≥ 1. In addition, we present
some examples to illustrate the proposed theories. Moreover, we show that some families of graphs admit
odd harmonious libeling.
This document provides an introduction to graph theory concepts. It defines what a graph is consisting of vertices and edges. It discusses different types of graphs like simple graphs, multigraphs, digraphs and their properties. It introduces concepts like degrees of vertices, handshaking lemma, planar graphs, Euler's formula, bipartite graphs and graph coloring. It provides examples of special graphs like complete graphs, cycles, wheels and hypercubes. It discusses applications of graphs in areas like job assignments and local area networks. The document also summarizes theorems regarding planar graphs like Kuratowski's theorem stating conditions for a graph to be non-planar.
Some Concepts on Constant Interval Valued Intuitionistic Fuzzy Graphsiosrjce
IOSR Journal of Mathematics(IOSR-JM) is a double blind peer reviewed International Journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
This document contains definitions and explanations of various graph theory terms in the form of 26 multiple choice questions. Some key terms defined and explained include: graph, simple graph, connected graph, components, tree, rooted tree, binary tree, walk, path, circuit, degree, adjacency, incidence, isomorphism, subgraph, Euler graph, and Hamiltonian path. Examples are provided to illustrate many of the graph theory concepts discussed.
This document discusses operations on interval-valued fuzzy graphs. It begins with an introduction to fuzzy graphs and definitions related to fuzzy graph theory. It then presents the main results which prove properties of the union and join operations on interval-valued fuzzy graphs. Specifically, it proves that the union of two interval-valued fuzzy graphs G1 and G2 is isomorphic to their join, and vice versa. It also discusses subgraphs, complements, and neighbors in fuzzy graph theory.
The document provides definitions and examples related to graphs. It defines graphs, directed graphs, and terminology like vertices, edges, degrees, adjacency, and isomorphism. It discusses representations of graphs using adjacency lists and matrices. It also covers special types of graphs like trees, cycles, and bipartite graphs. Key concepts are illustrated with examples.
New Classes of Odd Graceful Graphs - M. E. Abdel-AalGiselleginaGloria
In this paper, we introduce the notions of m-shadow graphs and n-splitting graphs,m ≥ 2, n ≥ 1. We
prove that, the m-shadow graphs for paths, complete bipartite graphs and symmetric product between
paths and null graphs are odd graceful. In addition, we show that, the m-splitting graphs for paths, stars
and symmetric product between paths and null graphs are odd graceful. Finally, we present some examples
to illustrate the proposed theories.
The document defines graphs and discusses different types of graphs and graph representations. It begins by defining what a graph is - a set of vertices and edges. It describes directed vs undirected graphs. It then discusses different graph representations including adjacency matrices and adjacency lists. It also covers graph operations and concepts such as degree, traversal methods like depth-first search and breadth-first search, connected components, spanning trees, and minimum cost spanning trees.
Graphs are propular to visualize a problem . Matrix representation is use to convert the graph in a form that used by the computer . This will help to get the efficent solution also provide a lots of mathematical equation .
Graph theory concepts can be applied in various domains such as computer networks, chemistry, and transportation. Key graph theory concepts include:
- Graphs are represented by vertices (nodes) and edges, and can be directed or undirected. Common graph types include trees, cycles, and bipartite graphs.
- Graphs can be represented through matrices like the incidence matrix and adjacency matrix which describe the relationships between vertices and edges. Adjacency lists are another representation.
- Connectivity describes the reachability between vertices via paths along edges. A graph is connected if there is a path between all pairs of vertices.
- Other concepts include isomorphism, which determines if two graphs have the same structure, and
This document defines basic concepts in graph theory. A graph consists of a set of vertices and edges connecting pairs of vertices. An adjacency matrix represents which vertices are connected by edges, while an incidence matrix represents which edges connect to each vertex. A simple graph cannot have loops or multiple edges between vertices. A complete graph connects each pair of vertices. Two graphs are isomorphic if there is a one-to-one correspondence between their vertices that preserves edge connections. Multigraphs allow multiple edges between vertices, while pseudographs also allow loops. The degree of a vertex is the number of edges connected to it.
Total Dominating Color Transversal Number of Graphs And Graph Operationsinventionjournals
Total Dominating Color Transversal Set of a graph is a Total Dominating Set of the graph which is also Transversal of Some 휒 - Partition of the graph. Here 휒 is the Chromatic number of the graph. Total Dominating Color Transversal number of a graph is the cardinality of a Total Dominating Color Transversal Set which has minimum cardinality among all such sets that the graph admits. In this paper, we consider the well known graph operations Join, Corona, Strong product and Lexicographic product of graphs and determine Total Dominating Color Transversal number of the resultant graphs.
This document defines and describes concepts related to fuzzy graphs and fuzzy digraphs. Key points include:
- A fuzzy graph is defined by two functions that assign membership values to vertices and edges.
- A fuzzy subgraph has lower or equal membership values for vertices and edges compared to the original graph.
- Effective edges and effective paths only include edges/paths where the membership equals the minimum vertex membership.
- Various graph measures are generalized to fuzzy graphs, such as vertex degree, order, size, and domination number.
- Fuzzy digraphs are defined similarly but with directed edges. Concepts like paths, independence, and domination are extended to fuzzy digraphs.
The document summarizes research characterizing graphs with specific relationships between their chromatic number (χ), domination number (γ), and complementary connected domination number (γcc). It is shown that a graph has γcc = χ = 2 if and only if it is isomorphic to a graph formed by adding two vertices to a bipartite graph. For r-regular graphs, γcc = χ = 2 if and only if the graph is isomorphic to either Kr,r or Kr+1,r+1 minus a matching.
This document defines and provides examples of graphs and graph representations. It begins by discussing Euler's use of graphs to solve the Königsberg bridge problem. It then defines graphs formally as G=(V,E) with vertices V and edges E. Examples of undirected and directed graphs are given. Common graph terms like paths, cycles, and connectivity are defined. Methods of representing graphs through adjacency matrices and adjacency lists are described and examples are provided. Finally, the document briefly discusses graph traversal algorithms and the concept of weighted edges.
Similar to On Embedding and NP-Complete Problems of Equitable Labelings (18)
This document provides a technical review of secure banking using RSA and AES encryption methodologies. It discusses how RSA and AES are commonly used encryption standards for secure data transmission between ATMs and bank servers. The document first provides background on ATM security measures and risks of attacks. It then reviews related work analyzing encryption techniques. The document proposes using a one-time password in addition to a PIN for ATM authentication. It concludes that implementing encryption standards like RSA and AES can make transactions more secure and build trust in online banking.
This document analyzes the performance of various modulation schemes for achieving energy efficient communication over fading channels in wireless sensor networks. It finds that for long transmission distances, low-order modulations like BPSK are optimal due to their lower SNR requirements. However, as transmission distance decreases, higher-order modulations like 16-QAM and 64-QAM become more optimal since they can transmit more bits per symbol, outweighing their higher SNR needs. Simulations show lifetime extensions up to 550% are possible in short-range networks by using higher-order modulations instead of just BPSK. The optimal modulation depends on transmission distance and balancing the energy used by electronic components versus power amplifiers.
This document provides a review of mobility management techniques in vehicular ad hoc networks (VANETs). It discusses three modes of communication in VANETs: vehicle-to-infrastructure (V2I), vehicle-to-vehicle (V2V), and hybrid vehicle (HV) communication. For each communication mode, different mobility management schemes are required due to their unique characteristics. The document also discusses mobility management challenges in VANETs and outlines some open research issues in improving mobility management for seamless communication in these dynamic networks.
This document provides a review of different techniques for segmenting brain MRI images to detect tumors. It compares the K-means and Fuzzy C-means clustering algorithms. K-means is an exclusive clustering algorithm that groups data points into distinct clusters, while Fuzzy C-means is an overlapping clustering algorithm that allows data points to belong to multiple clusters. The document finds that Fuzzy C-means requires more time for brain tumor detection compared to other methods like hierarchical clustering or K-means. It also reviews related work applying these clustering algorithms to segment brain MRI images.
1) The document simulates and compares the performance of AODV and DSDV routing protocols in a mobile ad hoc network under three conditions: when users are fixed, when users move towards the base station, and when users move away from the base station.
2) The results show that both protocols have higher packet delivery and lower packet loss when users are either fixed or moving towards the base station, since signal strength is better in those scenarios. Performance degrades when users move away from the base station due to weaker signals.
3) AODV generally has better performance than DSDV, with higher throughput and packet delivery rates observed across the different user mobility conditions.
This document describes the design and implementation of 4-bit QPSK and 256-bit QAM modulation techniques using MATLAB. It compares the two techniques based on SNR, BER, and efficiency. The key steps of implementing each technique in MATLAB are outlined, including generating random bits, modulation, adding noise, and measuring BER. Simulation results show scatter plots and eye diagrams of the modulated signals. A table compares the results, showing that 256-bit QAM provides better performance than 4-bit QPSK. The document concludes that QAM modulation is more effective for digital transmission systems.
The document proposes a hybrid technique using Anisotropic Scale Invariant Feature Transform (A-SIFT) and Robust Ensemble Support Vector Machine (RESVM) to accurately identify faces in images. A-SIFT improves upon traditional SIFT by applying anisotropic scaling to extract richer directional keypoints. Keypoints are processed with RESVM and hypothesis testing to increase accuracy above 95% by repeatedly reprocessing images until the threshold is met. The technique was tested on similar and different facial images and achieved better results than SIFT in retrieval time and reduced keypoints.
This document studies the effects of dielectric superstrate thickness on microstrip patch antenna parameters. Three types of probes-fed patch antennas (rectangular, circular, and square) were designed to operate at 2.4 GHz using Arlondiclad 880 substrate. The antennas were tested with and without an Arlondiclad 880 superstrate of varying thicknesses. It was found that adding a superstrate slightly degraded performance by lowering the resonant frequency and increasing return loss and VSWR, while decreasing bandwidth and gain. Specifically, increasing the superstrate thickness or dielectric constant resulted in greater changes to the antenna parameters.
This document describes a wireless environment monitoring system that utilizes soil energy as a sustainable power source for wireless sensors. The system uses a microbial fuel cell to generate electricity from the microbial activity in soil. Two microbial fuel cells were created using different soil types and various additives to produce different current and voltage outputs. An electronic circuit was designed on a printed circuit board with components like a microcontroller and ZigBee transceiver. Sensors for temperature and humidity were connected to the circuit to monitor the environment wirelessly. The system provides a low-cost way to power remote sensors without needing battery replacement and avoids the high costs of wiring a power source.
1) The document proposes a model for a frequency tunable inverted-F antenna that uses ferrite material.
2) The resonant frequency of the antenna can be significantly shifted from 2.41GHz to 3.15GHz, a 31% shift, by increasing the static magnetic field placed on the ferrite material.
3) Altering the permeability of the ferrite allows tuning of the antenna's resonant frequency without changing the physical dimensions, providing flexibility to operate over a wide frequency range.
This document summarizes a research paper that presents a speech enhancement method using stationary wavelet transform. The method first classifies speech into voiced, unvoiced, and silence regions based on short-time energy. It then applies different thresholding techniques to the wavelet coefficients of each region - modified hard thresholding for voiced speech, semi-soft thresholding for unvoiced speech, and setting coefficients to zero for silence. Experimental results using speech from the TIMIT database corrupted with white Gaussian noise at various SNR levels show improved performance over other popular denoising methods.
This document reviews the design of an energy-optimized wireless sensor node that encrypts data for transmission. It discusses how sensing schemes that group nodes into clusters and transmit aggregated data can reduce energy consumption compared to individual node transmissions. The proposed node design calculates the minimum transmission power needed based on received signal strength and uses a periodic sleep/wake cycle to optimize energy when not sensing or transmitting. It aims to encrypt data at both the node and network level to further optimize energy usage for wireless communication.
This document discusses group consumption modes. It analyzes factors that impact group consumption, including external environmental factors like technological developments enabling new forms of online and offline interactions, as well as internal motivational factors at both the group and individual level. The document then proposes that group consumption modes can be divided into four types based on two dimensions: vertical (group relationship intensity) and horizontal (consumption action period). These four types are instrument-oriented, information-oriented, enjoyment-oriented, and relationship-oriented consumption modes. Finally, the document notes that consumption modes are dynamic and can evolve over time.
The document summarizes a study of different microstrip patch antenna configurations with slotted ground planes. Three antenna designs were proposed and their performance evaluated through simulation: a conventional square patch, an elliptical patch, and a star-shaped patch. All antennas were mounted on an FR4 substrate. The effects of adding different slot patterns to the ground plane on resonance frequency, bandwidth, gain and efficiency were analyzed parametrically. Key findings were that reshaping the patch and adding slots increased bandwidth and shifted resonance frequency. The elliptical and star patches in particular performed better than the conventional design. Three antenna configurations were selected for fabrication and measurement based on the simulations: a conventional patch with a slot under the patch, an elliptical patch with slots
1) The document describes a study conducted to improve call drop rates in a GSM network through RF optimization.
2) Drive testing was performed before and after optimization using TEMS software to record network parameters like RxLevel, RxQuality, and events.
3) Analysis found call drops were occurring due to issues like handover failures between sectors, interference from adjacent channels, and overshooting due to antenna tilt.
4) Corrective actions taken included defining neighbors between sectors, adjusting frequencies to reduce interference, and lowering the mechanical tilt of an antenna.
5) Post-optimization drive testing showed improvements in RxLevel, RxQuality, and a reduction in dropped calls.
This document describes the design of an intelligent autonomous wheeled robot that uses RF transmission for communication. The robot has two modes - automatic mode where it can make its own decisions, and user control mode where a user can control it remotely. It is designed using a microcontroller and can perform tasks like object recognition using computer vision and color detection in MATLAB, as well as wall painting using pneumatic systems. The robot's movement is controlled by DC motors and it uses sensors like ultrasonic sensors and gas sensors to navigate autonomously. RF transmission allows communication between the robot and a remote control unit. The overall aim is to develop a low-cost robotic system for industrial applications like material handling.
This document reviews cryptography techniques to secure the Ad-hoc On-Demand Distance Vector (AODV) routing protocol in mobile ad-hoc networks. It discusses various types of attacks on AODV like impersonation, denial of service, eavesdropping, black hole attacks, wormhole attacks, and Sybil attacks. It then proposes using the RC6 cryptography algorithm to secure AODV by encrypting data packets and detecting and removing malicious nodes launching black hole attacks. Simulation results show that after applying RC6, the packet delivery ratio and throughput of AODV increase while delay decreases, improving the security and performance of the network under attack.
The document describes a proposed modification to the conventional Booth multiplier that aims to increase its speed by applying concepts from Vedic mathematics. Specifically, it utilizes the Urdhva Tiryakbhyam formula to generate all partial products concurrently rather than sequentially. The proposed 8x8 bit multiplier was coded in VHDL, simulated, and found to have a path delay 44.35% lower than a conventional Booth multiplier, demonstrating its potential for higher speed.
This document discusses image deblurring techniques. It begins by introducing image restoration and focusing on image deblurring. It then discusses challenges with image deblurring being an ill-posed problem. It reviews existing approaches to screen image deconvolution including estimating point spread functions and iteratively estimating blur kernels and sharp images. The document also discusses handling spatially variant blur and summarizes the relationship between the proposed method and previous work for different blur types. It proposes using color filters in the aperture to exploit parallax cues for segmentation and blur estimation. Finally, it proposes moving the image sensor circularly during exposure to prevent high frequency attenuation from motion blur.
This document describes modeling an adaptive controller for an aircraft roll control system using PID, fuzzy-PID, and genetic algorithm. It begins by introducing the aircraft roll control system and motivation for developing an adaptive controller to minimize errors from noisy analog sensor signals. It then provides the mathematical model of aircraft roll dynamics and describes modeling the real-time flight control system in MATLAB/Simulink. The document evaluates PID, fuzzy-PID, and PID-GA (genetic algorithm) controllers for aircraft roll control and finds that the PID-GA controller delivers the best performance.
The binding of cosmological structures by massless topological defectsSérgio Sacani
Assuming spherical symmetry and weak field, it is shown that if one solves the Poisson equation or the Einstein field
equations sourced by a topological defect, i.e. a singularity of a very specific form, the result is a localized gravitational
field capable of driving flat rotation (i.e. Keplerian circular orbits at a constant speed for all radii) of test masses on a thin
spherical shell without any underlying mass. Moreover, a large-scale structure which exploits this solution by assembling
concentrically a number of such topological defects can establish a flat stellar or galactic rotation curve, and can also deflect
light in the same manner as an equipotential (isothermal) sphere. Thus, the need for dark matter or modified gravity theory is
mitigated, at least in part.
Or: Beyond linear.
Abstract: Equivariant neural networks are neural networks that incorporate symmetries. The nonlinear activation functions in these networks result in interesting nonlinear equivariant maps between simple representations, and motivate the key player of this talk: piecewise linear representation theory.
Disclaimer: No one is perfect, so please mind that there might be mistakes and typos.
dtubbenhauer@gmail.com
Corrected slides: dtubbenhauer.com/talks.html
Phenomics assisted breeding in crop improvementIshaGoswami9
As the population is increasing and will reach about 9 billion upto 2050. Also due to climate change, it is difficult to meet the food requirement of such a large population. Facing the challenges presented by resource shortages, climate
change, and increasing global population, crop yield and quality need to be improved in a sustainable way over the coming decades. Genetic improvement by breeding is the best way to increase crop productivity. With the rapid progression of functional
genomics, an increasing number of crop genomes have been sequenced and dozens of genes influencing key agronomic traits have been identified. However, current genome sequence information has not been adequately exploited for understanding
the complex characteristics of multiple gene, owing to a lack of crop phenotypic data. Efficient, automatic, and accurate technologies and platforms that can capture phenotypic data that can
be linked to genomics information for crop improvement at all growth stages have become as important as genotyping. Thus,
high-throughput phenotyping has become the major bottleneck restricting crop breeding. Plant phenomics has been defined as the high-throughput, accurate acquisition and analysis of multi-dimensional phenotypes
during crop growing stages at the organism level, including the cell, tissue, organ, individual plant, plot, and field levels. With the rapid development of novel sensors, imaging technology,
and analysis methods, numerous infrastructure platforms have been developed for phenotyping.
Describing and Interpreting an Immersive Learning Case with the Immersion Cub...Leonel Morgado
Current descriptions of immersive learning cases are often difficult or impossible to compare. This is due to a myriad of different options on what details to include, which aspects are relevant, and on the descriptive approaches employed. Also, these aspects often combine very specific details with more general guidelines or indicate intents and rationales without clarifying their implementation. In this paper we provide a method to describe immersive learning cases that is structured to enable comparisons, yet flexible enough to allow researchers and practitioners to decide which aspects to include. This method leverages a taxonomy that classifies educational aspects at three levels (uses, practices, and strategies) and then utilizes two frameworks, the Immersive Learning Brain and the Immersion Cube, to enable a structured description and interpretation of immersive learning cases. The method is then demonstrated on a published immersive learning case on training for wind turbine maintenance using virtual reality. Applying the method results in a structured artifact, the Immersive Learning Case Sheet, that tags the case with its proximal uses, practices, and strategies, and refines the free text case description to ensure that matching details are included. This contribution is thus a case description method in support of future comparative research of immersive learning cases. We then discuss how the resulting description and interpretation can be leveraged to change immersion learning cases, by enriching them (considering low-effort changes or additions) or innovating (exploring more challenging avenues of transformation). The method holds significant promise to support better-grounded research in immersive learning.
EWOCS-I: The catalog of X-ray sources in Westerlund 1 from the Extended Weste...Sérgio Sacani
Context. With a mass exceeding several 104 M⊙ and a rich and dense population of massive stars, supermassive young star clusters
represent the most massive star-forming environment that is dominated by the feedback from massive stars and gravitational interactions
among stars.
Aims. In this paper we present the Extended Westerlund 1 and 2 Open Clusters Survey (EWOCS) project, which aims to investigate
the influence of the starburst environment on the formation of stars and planets, and on the evolution of both low and high mass stars.
The primary targets of this project are Westerlund 1 and 2, the closest supermassive star clusters to the Sun.
Methods. The project is based primarily on recent observations conducted with the Chandra and JWST observatories. Specifically,
the Chandra survey of Westerlund 1 consists of 36 new ACIS-I observations, nearly co-pointed, for a total exposure time of 1 Msec.
Additionally, we included 8 archival Chandra/ACIS-S observations. This paper presents the resulting catalog of X-ray sources within
and around Westerlund 1. Sources were detected by combining various existing methods, and photon extraction and source validation
were carried out using the ACIS-Extract software.
Results. The EWOCS X-ray catalog comprises 5963 validated sources out of the 9420 initially provided to ACIS-Extract, reaching a
photon flux threshold of approximately 2 × 10−8 photons cm−2
s
−1
. The X-ray sources exhibit a highly concentrated spatial distribution,
with 1075 sources located within the central 1 arcmin. We have successfully detected X-ray emissions from 126 out of the 166 known
massive stars of the cluster, and we have collected over 71 000 photons from the magnetar CXO J164710.20-455217.
When I was asked to give a companion lecture in support of ‘The Philosophy of Science’ (https://shorturl.at/4pUXz) I decided not to walk through the detail of the many methodologies in order of use. Instead, I chose to employ a long standing, and ongoing, scientific development as an exemplar. And so, I chose the ever evolving story of Thermodynamics as a scientific investigation at its best.
Conducted over a period of >200 years, Thermodynamics R&D, and application, benefitted from the highest levels of professionalism, collaboration, and technical thoroughness. New layers of application, methodology, and practice were made possible by the progressive advance of technology. In turn, this has seen measurement and modelling accuracy continually improved at a micro and macro level.
Perhaps most importantly, Thermodynamics rapidly became a primary tool in the advance of applied science/engineering/technology, spanning micro-tech, to aerospace and cosmology. I can think of no better a story to illustrate the breadth of scientific methodologies and applications at their best.
ESPP presentation to EU Waste Water Network, 4th June 2024 “EU policies driving nutrient removal and recycling
and the revised UWWTD (Urban Waste Water Treatment Directive)”
hematic appreciation test is a psychological assessment tool used to measure an individual's appreciation and understanding of specific themes or topics. This test helps to evaluate an individual's ability to connect different ideas and concepts within a given theme, as well as their overall comprehension and interpretation skills. The results of the test can provide valuable insights into an individual's cognitive abilities, creativity, and critical thinking skills
Remote Sensing and Computational, Evolutionary, Supercomputing, and Intellige...University of Maribor
Slides from talk:
Aleš Zamuda: Remote Sensing and Computational, Evolutionary, Supercomputing, and Intelligent Systems.
11th International Conference on Electrical, Electronics and Computer Engineering (IcETRAN), Niš, 3-6 June 2024
Inter-Society Networking Panel GRSS/MTT-S/CIS Panel Session: Promoting Connection and Cooperation
https://www.etran.rs/2024/en/home-english/
Current Ms word generated power point presentation covers major details about the micronuclei test. It's significance and assays to conduct it. It is used to detect the micronuclei formation inside the cells of nearly every multicellular organism. It's formation takes place during chromosomal sepration at metaphase.
On Embedding and NP-Complete Problems of Equitable Labelings
1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 11, Issue 1 Ver. III (Jan - Feb. 2015), PP 80-85
www.iosrjournals.org
DOI: 10.9790/5728-11138085 www.iosrjournals.org 80 |Page
On Embedding and NP-Complete Problems of Equitable
Labelings
S. K. Vaidya1
, C. M. Barasara2
1
(Department of Mathematics, Saurashtra University, Rajkot - 360 005, Gujarat, INDIA)
2
(Atmiya Institute of Technology and Science, Rajkot - 360 005, Gujarat, INDIA)
Abstract: The embedding and NP-complete problems in the context of various graph labeling schemes were
remained in the focus of many researchers. Here we have explored embedding and NP-Complete problems for
some variants of cordial labeling.
Keywords: Graph labeling, equitable labeling, embedding of graphs, NP-complete problems.
Mathematics Subject Classification (2010): 05C78, 05C60.
I. Introduction
A graph labeling is an assignment of integers to the vertices or edges, or both, subject to certain
condition or conditions.
Labeling of discrete structures is a potential area of research and a comprehensive survey on graph
labeling can be found in Gallian [1].
Definition 1.1: For a graph G , a vertex labeling function }1,0{)(: GVf induces an edge labeling function
}1,0{)(:*
GEf defined as )()()(*
vfufuvef . Then f is called cordial labeling of graph G if
absolute difference of number of vertices with label 1 and label 0 is at most 1 and absolute difference of number
of edges with label 1 and label 0 is at most 1.
A graph G is called cordial if it admits a cordial labeling.
The concept of cordial labeling was introduced by Cahit [2] in 1987 and in the same paper he presented
several results on this newly introduced concept.
After this some labelings like prime cordial labeling, A - cordial labeling, product cordial labeling,
edge product cordial labeling etc. were also introduced with minor variations in cordial theme. Such labelings
are classified as equitable labelings.
Definition 1.2: For a graph G , an edge labeling function }1,0{)(: GEf induces a vertex labeling function
}1,0{)(:*
GVf defined as )}(/)({)(*
GEuvuvfuf . Then f is called an edge product cordial labeling
of graph G if the absolute difference of number of vertices with label 1 and label 0 is at most 1 and the absolute
difference of number of edges with label 1 and label 0 is also at most 1.
A graph G is called edge product cordial graph if it admits an edge product cordial labeling.
In 2012, the concept of edge product cordial labeling was introduced by Vaidya and Barasara [3]. In
the same paper they have investigated edge product cordial labeling for some standard graphs.
For any graph G , we introduce following notations:
1. )1(fv = the number of vertices having label 1;
2. )0(fv = the number of vertices having label 0;
3. (1)fe = the number of edges having label 1;
4. (0)fe = the number of edges having label 0;
Throughout this discussion we consider simple, finite and undirected graph ))(),(( GEGVG with
pGV )( and qGE )( . For all other standard terminology and notation we refer to Chartrand and
Lesniak[4].
2. On Embedding and NP-Complete Problems of Equitable Labelings
DOI: 10.9790/5728-11138085 www.iosrjournals.org 81 |Page
II. Embedding of Equitable graphs
Definition 2.1: For a graphG , an edge labeling function }1,0{)(: GEf induces a vertex labeling function
}1,0{)(:*
GVf defined as )}(/)({)(*
GEuvuvfuf . Then f is called a total edge product cordial
labeling of graph G if (1) (1) (0) (0) 1f f f fv e v e . A graph G is called total edge product cordial
graph if it admits a total edge product cordial labeling.
The concept of total edge product cordial labeling was introduced by Vaidya and Barasara [5].
Theorem 2.2: Any graph G can be embedded as an induced subgraph of a total edge product cordial graph.
Proof. Label the edges of G in such a way that edges with label 1 and edges with label 0 differ by at most 1.
Let iV be the set of vertices with label i and iE be the set of edges with label i while in V and
in E be the cardinality of set iV and iE respectively.
Case 1: When 1 1 0 0( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 0V . Assign label 0 to these new edges. Consequently
2
r
vertices will receive label 0.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) ( ) ( ) 2 1
2
r
n V n E n V n E
.
Case 2: When 0 0 1 1( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 0V . Assign label 1 to these new edges. Consequently
2
r
vertices will receive label 1.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) 2 ( ) ( ) 1
2
r
n V n E n V n E
.
Thus in all the possibilities, the constructed supergraph H satisfies the conditions for total edge
product cordial graph. That is, any graph G can be embedded as an induced subgraph of a total edge product
cordial graph.
Definition 2.3: Let : ( ) {1,2,..., }f V G p be a bijection. For each edge uv , assign the label ( ) ( )f u f v .
Function f is called a difference cordial labeling if (1) (0) 1f fe e . where (1)fe and (0)fe denote the
number of edges labeled with 1 and not labeled with 1 respectively. A graph with a difference cordial labeling is
called a difference cordial graph.
The concept of difference cordial labeling was introduced by Ponraj et al. [6] in 2003.
Theorem 2.4: Any graph G can be embedded as an induced subgraph of a difference cordial graph.
Proof. Let G be the given graph with p vertices and q edges. As per definition of difference cordial labeling
assign labels {1,2,..., }p to the vertices of G .
Let iE be the set of edges with label i while in E be the cardinality of set iE .
Case 1: When 1 0( ) ( ) 1n E n E (say r ).
The new graph H can be obtained by adding r vertices, say 1 2{ , ,..., }rv v v to the graph G and join
them to the vertex with label 1. Assign label p i to the vertex iv for 1,2,...,i r . Consequently r edges will
receive label 0.
Hence, (1) (0)f fe e = 1 0( ) ( ) 0n E n E r .
Case 2: When 0 1( ) ( ) 1n E n E (say r ).
3. On Embedding and NP-Complete Problems of Equitable Labelings
DOI: 10.9790/5728-11138085 www.iosrjournals.org 82 |Page
The new graph H can be obtained by adding a path of r vertices, say 1 2{ , ,..., }rv v v to the graph G
and join the vertex 1v to the vertex with label p . Assign label p i to the vertex iv for 1,2,...,i r .
Consequently r edges will receive label 1.
Hence, (1) (0)f fe e = 1 0( ) ( ) 0n E r n E .
Thus in all the possibilities, the constructed supergraph H satisfies the conditions for difference
cordial graph. That is, any graph G can be embedded as an induced subgraph of a difference cordial graph.
Definition 2.5: A graph G is called simply sequential if there is a bijection : ( ) ( ) {1,2,..., }f V G E G p q
such that for each edge { } ( )e uv E G one has ( ) ( ) ( )f e f u f v .
In 1983, Bange et al. [7] have introduced the concept of sequential labeling.
Definition 2.6: If a mapping : ( ) ( ) {0,1}f V G E G is such that for each edge e uv , ( ) ( ) ( )f e f u f v
and the condition (1) (1) (0) (0) 1f f f fv e v e holds then G is called total sequential cordial graph.
In 2002, Cahit [8] has introduced the concept of total sequential cordial labeling as a weaker version of
sequential labeling.
Theorem 2.7: Any graph G can be embedded as an induced subgraph of a total sequential cordial graph.
Proof. Label the vertices of graph G in such a way that vertices with label 0 and vertices with label 1 differ by
at most 1.
Let iV be the set of vertices with label i and iE be the set of edges with label i while in V and
in E be the cardinality of set iV and iE respectively.
Case 1: When 1 1 0 0( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 0V . Assign label 0 to these new vertices. Consequently
2
r
edges will receive label 0.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) ( ) ( ) 2 1
2
r
n V n E n V n E
.
Case 2: When 0 0 1 1( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 0V . Assign label 1 to these new vertices. Consequently
2
r
edges will receive label 1.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) 2 ( ) ( ) 1
2
r
n V n E n V n E
.
Thus in all the possibilities, the constructed supergraph H satisfies the conditions for total sequential
cordial graph. That is, any graph G can be embedded as an induced subgraph of a total sequential cordial graph.
Definition 2.8: Let : ( ) {1,2,..., }f V G p be a bijection. For each edge uv , assign the label 1 if either
( ) | ( )f u f v or ( ) | ( )f v f u and the label 0 otherwise. f is called a divisor cordial labeling if (1) (0) 1f fe e .
A graph with a divisor cordial labeling is called a divisor cordial graph.
In 2011, Varatharajan et al. [9] have introduced the concept of divisor cordial labeling.
Theorem 2.9: Any graph G can be embedded as an induced subgraph of a divisor cordial graph.
Proof. Let G be the given graph with p vertices and q edges. As per definition of divisor cordial labeling
assign labels {1,2,..., }p to the vertices of G .
Let iE be the set of edges with label i while in E be the cardinality of set iE .
Case 1: When 0 1 1n E n E (say r ).
4. On Embedding and NP-Complete Problems of Equitable Labelings
DOI: 10.9790/5728-11138085 www.iosrjournals.org 83 |Page
The new graph H can be obtained by adding r vertices to the graph G , say 1 2, ,..., rv v v and join them
to vertex with label 1. Assign label p i to the vertex iv for 1,2,...,i r . Consequently r edges will receive
label 1.
Hence, (1) (0)f fe e = 0 1 0n E n E r .
Case 2: When 1 0 1n E n E (say r ).
The new graph H can be obtained by adding a path of r vertices, say 1 2, ,..., rv v v to the graph G and
join the vertex 1v to the vertex with label p . Assign label p i to the vertex iv for 1,2,...,i r .
Consequently r edges will receive label 0.
Hence, (1) (0)f fe e = 0 1 0n E r n E .
Thus in all the possibilities, the constructed supergraph H satisfies the conditions for divisor cordial
graph. That is, any graph G can be embedded as an induced subgraph of a divisor cordial graph.
Definition 2.10: A graph G is said to have a magic labeling with constant C if there exists a bijection
: ( ) ( ) {1,2,..., }f V G E G p q such that ( ) ( ) ( )f u f v f uv C for all ( )uv E G .
Kotzig and Rosa [10] have introduced the concept of magic labeling in 1970 and investigated magic
labeling for some standard graphs.
Definition 2.11: A graph G is said to have a totally magic cordial labeling with constant C if there exists a
mapping : ( ) ( ) {0,1}f V G E G such that ( ) ( ) ( )f u f v f uv C (mod 2) for all ( )uv E G provided the
condition (1) (1) (0) (0) 1f f f fv e v e holds.
In 2002, Cahit [8] has introduced the concept of total magic cordial labeling as a weaker version of
magic labeling.
Theorem 2.12: Any graph G can be embedded as an induced subgraph of a total magic cordial graph.
Proof. Label the vertices of graph G in such a way that (1) (0) 1f fv v and as per definition of total magic
cordial labeling assign edge labels such that ( ) ( ) ( )f u f v f uv C (mod 2).
Let iV be the set of vertices with label i and iE be the set of edges with label i while in V and
in E be the cardinality of set iV and iE respectively.
Case 1: If 0C .
Subcase 1: When 1 1 0 0( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 0V . Assign label 0 to these new vertices. Consequently
2
r
edges will receive label 0.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) ( ) ( ) 2 1
2
r
n V n E n V n E
.
Subcase 2: When 0 0 1 1( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 0V . Assign label 1 to these new vertices. Consequently
2
r
edges will receive label 1.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) 2 ( ) ( ) 1
2
r
n V n E n V n E
.
Case 2: If 1C .
Subcase 1: When 1 1 0 0( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
5. On Embedding and NP-Complete Problems of Equitable Labelings
DOI: 10.9790/5728-11138085 www.iosrjournals.org 84 |Page
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 1V . Assign label 0 to these new vertices. Consequently
2
r
edges will receive label 0.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) ( ) ( ) 2 1
2
r
n V n E n V n E
.
Subcase 2: When 0 0 1 1( ) ( ) ( ) ( ) 1n V n E n V n E (say r ).
The new graph H can be obtained by adding
2
r
vertices to the graph G and join them to arbitrary
member of 1V . Assign label 1 to these new vertices. Consequently
2
r
edges will receive label 1.
Thus, (1) (1) (0) (0)f f f fv e v e = 1 1 0 0( ) ( ) 2 ( ) ( ) 1
2
r
n V n E n V n E
.
Thus in all the possibilities, the constructed supergraph H satisfies the conditions for total magic
cordial graph. That is, any graph G can be embedded as an induced subgraph of a total magic cordial graph.
III. NP-Complete Problems
Theorem 3.1: Any planar graph G can be embedded as an induced subgraph of a planar total edge product
cordial (difference cordial, total sequential cordial, divisor cordial, total magic cordial) graph.
Proof. If G is planar graph. Then the supergraph H constructed in Theorem 2.2 (corresponding theorems 2.4,
2.7, 2.9 and 2.12) is a planar graph. Hence the result.
Theorem 3.2: Any connected graph G can be embedded as an induced subgraph of a connected total edge
product cordial (difference cordial, total sequential cordial, divisor cordial, total magic cordial) graph.
Proof. If G is connected graph. Then the supergraph H constructed in Theorem 2.2 (corresponding theorems
2.4, 2.7, 2.9 and 2.12) is a connected graph. Hence the result.
Theorem 3.3: The problem of deciding whether the chromatic number ( )G k , where 3k , is NP-complete
even for total edge product cordial (difference cordial, total sequential cordial, divisor cordial, total magic
cordial) graph.
Proof. Let G be a graph with chromatic number ( ) 3G . Let supergraph H constructed in Theorem 2.2 is
total edge product cordial (corresponding theorems 2.4, 2.7, 2.9 and 2.12), which contains G as an induced
subgraph. Then obviously we have ( ) ( )H H . Since the problem of deciding whether the chromatic
number ( )G k , where 3k , is NP-complete by [11]. It follows that deciding whether the chromatic
number ( )G k , where 3k , is NP-complete even for prime cordial graphs.
Theorem 3.4: The problem of deciding whether the clique number ( )G k is NP-complete even when
restricted to total edge product cordial (difference cordial, total sequential cordial, divisor cordial, total magic
cordial) graph.
Proof. Since the problem of deciding whether the clique number of a graph ( )G k is NP-complete by [11]
and ( ) ( )H G for the supergraph H constructed in Theorem 2.2 is total edge product cordial
(corresponding theorems 2.4, 2.7, 2.9 and 2.12). Hence the result.
Theorem 3.5: The problem of deciding whether the domination number (total domination number) is less than
or equal to k is NP-complete even when restricted to total edge product cordial (difference cordial, total
sequential cordial, divisor cordial, total magic cordial) graph.
Proof. Since the problem of deciding whether the domination number (total domination number) of a graph G
is less than or equal to k is NP-complete as reported in [11] and the supergraph H constructed in Theorem 2.2
is total edge product cordial (corresponding theorems 2.4, 2.7, 2.9 and 2.12) has domination number greater than
or equal to domination number of G . Hence the result.
6. On Embedding and NP-Complete Problems of Equitable Labelings
DOI: 10.9790/5728-11138085 www.iosrjournals.org 85 |Page
IV. Concluding Remarks
Acharya et al. [12] have proved that any graph G can be embedded as an induced subgraph of a
graceful graph. Thus showing the impossibility of obtaining any forbidden subgraph characterization for
graceful graphs on the same line Acharya et al. [13, 14], Anandavally et al. [15], Germina and Ajitha [16],
Vaidya and Vihol [17] and Vaidya and Barasara [18] have discussed embedding and NP-Complete problems in
the context of various graph labeling schemes. However here we have discussed embedding and NP-Complete
problems for total edge product cordial labeling, difference cordial labeling, total sequential cordial labeling,
divisor cordial labeling and total magic cordial labeling.
Acknowledgement
The authors are highly thankful to the anonymous referee for their kind suggestions and comments.
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