The idea of metric dimension in graph theory was introduced by P J Slater in [2]. It has been found
applications in optimization, navigation, network theory, image processing, pattern recognition etc.
Several other authors have studied metric dimension of various standard graphs. In this paper we
introduce a real valued function called generalized metric → + Gd
: X × X × X R where X = r(v /W) =
{(d(v,v1
),d(v,v2
),...,d(v,vk
/) v∈V (G))}, denoted Gd
and is used to study metric dimension of graphs. It
has been proved that metric dimension of any connected finite simple graph remains constant if Gd
numbers of pendant edges are added to the non-basis vertices.
1) The document discusses partial derivatives, which involve differentiating functions of two or more variables with respect to one variable while holding the others constant. It provides examples of computing first and second partial derivatives.
2) Implicit differentiation is introduced as a way to find partial derivatives of functions defined implicitly rather than explicitly. The chain rule is also discussed.
3) Methods are presented for finding partial derivatives of functions of two or three variables, including using implicit differentiation and the chain rule. Examples are provided to illustrate these concepts.
Solving second order ordinary differential equations (boundary value problems) using the Least Squares Technique. Contains one numerical examples from Shah, Eldho, Desai
This document provides a review worksheet on graphing quadratics. It includes definitions of key terms like standard form, parabola, axis of symmetry, vertex, minimum, maximum, zeros, and x-intercepts. It asks students to identify critical values like the axis of symmetry and vertex to graph quadratic functions. Students must determine the number of real roots, identify any real roots, and specify if the function has a minimum or maximum. The final problem asks students to analyze a quadratic model of a dolphin jump to find the maximum height and distance traveled.
Integration by Parts allows us to integrate some products using the formula:
dv du
u dx = uv - v dx
dx dx
This formula means that we differentiate one factor to get du/dx and integrate the other factor to get v. We then substitute these into the formula to evaluate the integral. For example, to integrate xsinx we would have u = x and dv/dx = sinx, then substitute into the formula. This technique can be used to evaluate integrals of products that cannot be integrated by other methods.
The document provides an introduction to graph theory. It lists prescribed and recommended books, outlines topics that will be covered including history, definitions, types of graphs, terminology, representation, subgraphs, connectivity, and applications. It notes that the Government of India designated June 10th as Graph Theory Day in recognition of the influence and importance of graph theory.
Dokumen tersebut membahas tentang sifat Archimedes dan beberapa teorema yang berkaitan dengan hubungan antara bilangan riil dan bilangan asli, termasuk bukti dari teorema bahwa terdapat bilangan riil positif x sedemikian sehingga x^2 = 2.
Teks tersebut membahas tentang persamaan linear orde tinggi dengan koefisien konstan. Ia menjelaskan bahwa solusi umum dari persamaan tersebut dapat ditentukan dengan menemukan akar dari persamaan karakteristik yang dihasilkan dari koefisien persamaan. Jika akar tersebut nyata dan berbeda, solusi umum berupa kombinasi fungsi eksponensial. Jika akarnya kompleks, solusi umum dapat ditulis menggunak
1) The document discusses partial derivatives, which involve differentiating functions of two or more variables with respect to one variable while holding the others constant. It provides examples of computing first and second partial derivatives.
2) Implicit differentiation is introduced as a way to find partial derivatives of functions defined implicitly rather than explicitly. The chain rule is also discussed.
3) Methods are presented for finding partial derivatives of functions of two or three variables, including using implicit differentiation and the chain rule. Examples are provided to illustrate these concepts.
Solving second order ordinary differential equations (boundary value problems) using the Least Squares Technique. Contains one numerical examples from Shah, Eldho, Desai
This document provides a review worksheet on graphing quadratics. It includes definitions of key terms like standard form, parabola, axis of symmetry, vertex, minimum, maximum, zeros, and x-intercepts. It asks students to identify critical values like the axis of symmetry and vertex to graph quadratic functions. Students must determine the number of real roots, identify any real roots, and specify if the function has a minimum or maximum. The final problem asks students to analyze a quadratic model of a dolphin jump to find the maximum height and distance traveled.
Integration by Parts allows us to integrate some products using the formula:
dv du
u dx = uv - v dx
dx dx
This formula means that we differentiate one factor to get du/dx and integrate the other factor to get v. We then substitute these into the formula to evaluate the integral. For example, to integrate xsinx we would have u = x and dv/dx = sinx, then substitute into the formula. This technique can be used to evaluate integrals of products that cannot be integrated by other methods.
The document provides an introduction to graph theory. It lists prescribed and recommended books, outlines topics that will be covered including history, definitions, types of graphs, terminology, representation, subgraphs, connectivity, and applications. It notes that the Government of India designated June 10th as Graph Theory Day in recognition of the influence and importance of graph theory.
Dokumen tersebut membahas tentang sifat Archimedes dan beberapa teorema yang berkaitan dengan hubungan antara bilangan riil dan bilangan asli, termasuk bukti dari teorema bahwa terdapat bilangan riil positif x sedemikian sehingga x^2 = 2.
Teks tersebut membahas tentang persamaan linear orde tinggi dengan koefisien konstan. Ia menjelaskan bahwa solusi umum dari persamaan tersebut dapat ditentukan dengan menemukan akar dari persamaan karakteristik yang dihasilkan dari koefisien persamaan. Jika akar tersebut nyata dan berbeda, solusi umum berupa kombinasi fungsi eksponensial. Jika akarnya kompleks, solusi umum dapat ditulis menggunak
This document provides an overview of integration and how it relates to calculating areas under curves and between functions. Some key points covered include:
- Integration uses the concept of reverse differentiation to calculate the area under a function.
- The area under a function between two x-values a and b is calculated as the definite integral from a to b.
- Integrals may be used to find the area between two curves by subtracting their integrals between the intersection points.
- Integrals can represent either positive or negative areas depending on whether the area is above or below the x-axis.
Solutions manual for analysis with an introduction to proof 5th edition by layteckli
Solutions Manual for Analysis With an Introduction to Proof 5th Edition by Lay
download at: https://goo.gl/fwifZP
people also search:
analysis with an introduction to proof 5th edition pdf
analysis with an introduction to proof 5th edition download
analysis with an introduction to proof 5th edition pdf free
analysis with an introduction to proof 5th edition by steven r lay pdf
analysis with an introduction to proof 4th edition
analysis with an introduction to proof 5th edition solutions manual pdf
analysis with an introduction to proof pdf download
analysis with an introduction to proof 5th edition solutions manual pdf free
This document provides an overview of integral calculus. It defines integration as the reverse process of differentiation and discusses indefinite and definite integrals. Graphical representations and general integration rules are presented. Examples are provided for integrals of simple functions using substitution and integration by parts methods. The document also covers integrals of trigonometric functions and derives formulas for several integrals. It concludes with examples of evaluating definite integrals between specified limits to find the area under a curve.
This document provides definitions and theorems related to graph theory. It begins with definitions of simple graphs, vertices, edges, degree, and the handshaking lemma. It then covers definitions and properties of paths, cycles, adjacency matrices, connectedness, Euler paths and circuits. The document also discusses Hamilton paths, planar graphs, trees, and other special types of graphs like complete graphs and bipartite graphs. It provides examples and proofs of many graph theory concepts and results.
Graph theory is the study of points and lines, and concepts like vertices and edges. It has many applications, including map coloring, Google Maps, and football games. Graph theory can be used to color maps with the minimum number of colors without any adjacent regions sharing the same color. It is also used in Google Maps for functions like estimating travel times and showing traffic conditions. In football, a match can be represented as a network with players as vertices and passes as edges, allowing analysis of things like which team has a more balanced network.
This document contains exercises involving functions and graphs. There are multiple choice and free response questions testing understanding of properties of functions including domain, range, intercepts, maxima/minima, and graphing functions. Solutions are provided for checking work.
This document contains exercises and solutions related to monotone sequences, convergent subsequences, and the Bolzano-Weierstrass theorem from an introduction to real analysis course. It includes 4 problems examining properties of specific sequences, showing whether they are bounded/monotone and finding their limits, as well as examples of an unbounded sequence with a convergent subsequence and sequences that diverge. The solutions provide detailed proofs of the properties of each sequence using induction and algebraic manipulations.
1. The document defines various types of graphs including simple graphs, multigraphs, pseudographs, directed graphs, and bipartite graphs. It discusses properties like vertices, edges, degrees, adjacency, and special graph structures like complete graphs, cycles, wheels, and n-cubes.
2. Key terms are defined, such as the degree of a vertex, adjacent vertices, and handshaking theorems for both undirected and directed graphs. Special graph structures including complete graphs, cycles, wheels, and n-cubes are also described.
3. Bipartite graphs are introduced as simple graphs whose vertices can be partitioned into two disjoint sets such that every edge connects a vertex in one set to a
1. Dokumen membahas penggunaan integral tentu untuk menghitung luas daerah yang dibatasi kurva dan volume benda putar
2. Ada beberapa metode untuk menghitung volume benda putar yaitu metode cakram, cincin, dan kulit tabung
3. Beberapa contoh soal diberikan beserta penyelesaiannya untuk mengilustrasikan konsep-konsep tersebut
Dokumen menjelaskan tentang kombinasi linier vektor. Kombinasi linier adalah penjumlahan vektor yang diperoleh dengan mengalikan suatu skalar dengan vektor. Rentang vektor adalah himpunan semua kombinasi linier dari vektor-vektor tersebut.
The document provides notes on group theory. It discusses the definition of groups and examples of groups such as (Z, +), (Q, ×), and Sn. Properties of groups like Lagrange's theorem and criteria for subgroups are also covered. The notes then discuss symmetry groups, defining isometries of R2 and showing that the set of isometries forms a group. Symmetry groups G(Π) of objects Π in R2 are introduced and shown to be subgroups. Specific examples of symmetry groups like those of triangles, squares, regular n-gons, and infinite strips are analyzed. Finally, the concept of group isomorphism is defined and examples are given to illustrate isomorphic groups.
Introduction of metric dimension of circular graphs is connected graph , The distance and diameter , Resolving sets and location number then Examples . Application in facility location problems . is has motivation (Applications in Chemistry and Networks systems). Definitions of Certain Regular Graphs. Main Results for three graphs (Prism , Antiprism and generalized Petersen graphs .
This document discusses edge-magic total labelings of graphs. Some key points:
- An edge-magic total labeling assigns integers to the vertices and edges of a graph such that the sum of each edge's label and the labels of its incident vertices is a constant value called the magic sum.
- Necessary conditions for a graph to have an edge-magic total labeling include distinct sums for edge labels and no repeated vertex label sums for non-adjacent vertices.
- Some elementary counting formulas are derived relating the vertex labels, degrees, magic sum, and number of edges and vertices.
- It is shown that if a graph has an even number of edges and the sum of its vertices and edges is
The document discusses different types of topological spaces, including Hausdorff and non-Hausdorff spaces. It defines separation axioms like T0, T1, T2, etc. and explains that metric spaces and the real number line are Hausdorff. Non-Hausdorff spaces are exemplified using an equivalence relation on the space [0,1]∪[2,3]. Regular, normal and compact Hausdorff spaces are also discussed.
The document discusses surface integrals. It defines a surface integral as integrating a density function w over a surface σ. The surface σ is defined by a function z=f(x,y) over a domain D. The surface is partitioned into small patches, and each patch's area is approximated. The total mass of the surface is calculated as the limit of Riemann sums as the partitions approach zero. An example calculates the mass of a density function over a spherical surface portion.
METRIC DIMENSION AND UNCERTAINTY OF TRAVERSING ROBOTS IN A NETWORKFransiskeran
Metric dimension in graph theory has many applications in the real world. It has been applied to the
optimization problems in complex networks, analyzing electrical networks; show the business relations,
robotics, control of production processes etc. This paper studies the metric dimension of graphs with
respect to contraction and its bijection between them. Also an algorithm to avoid the overlapping between
the robots in a network is introduced.
METRIC DIMENSION AND UNCERTAINTY OF TRAVERSING ROBOTS IN A NETWORKgraphhoc
Metric dimension in graph theory has many applications in the real world. It has been applied to the
optimization problems in complex networks, analyzing electrical networks; show the business relations,
robotics, control of production processes etc. This paper studies the metric dimension of graphs with
respect to contraction and its bijection between them. Also an algorithm to avoid the overlapping between
the robots in a network is introduced.
This document provides an overview of integration and how it relates to calculating areas under curves and between functions. Some key points covered include:
- Integration uses the concept of reverse differentiation to calculate the area under a function.
- The area under a function between two x-values a and b is calculated as the definite integral from a to b.
- Integrals may be used to find the area between two curves by subtracting their integrals between the intersection points.
- Integrals can represent either positive or negative areas depending on whether the area is above or below the x-axis.
Solutions manual for analysis with an introduction to proof 5th edition by layteckli
Solutions Manual for Analysis With an Introduction to Proof 5th Edition by Lay
download at: https://goo.gl/fwifZP
people also search:
analysis with an introduction to proof 5th edition pdf
analysis with an introduction to proof 5th edition download
analysis with an introduction to proof 5th edition pdf free
analysis with an introduction to proof 5th edition by steven r lay pdf
analysis with an introduction to proof 4th edition
analysis with an introduction to proof 5th edition solutions manual pdf
analysis with an introduction to proof pdf download
analysis with an introduction to proof 5th edition solutions manual pdf free
This document provides an overview of integral calculus. It defines integration as the reverse process of differentiation and discusses indefinite and definite integrals. Graphical representations and general integration rules are presented. Examples are provided for integrals of simple functions using substitution and integration by parts methods. The document also covers integrals of trigonometric functions and derives formulas for several integrals. It concludes with examples of evaluating definite integrals between specified limits to find the area under a curve.
This document provides definitions and theorems related to graph theory. It begins with definitions of simple graphs, vertices, edges, degree, and the handshaking lemma. It then covers definitions and properties of paths, cycles, adjacency matrices, connectedness, Euler paths and circuits. The document also discusses Hamilton paths, planar graphs, trees, and other special types of graphs like complete graphs and bipartite graphs. It provides examples and proofs of many graph theory concepts and results.
Graph theory is the study of points and lines, and concepts like vertices and edges. It has many applications, including map coloring, Google Maps, and football games. Graph theory can be used to color maps with the minimum number of colors without any adjacent regions sharing the same color. It is also used in Google Maps for functions like estimating travel times and showing traffic conditions. In football, a match can be represented as a network with players as vertices and passes as edges, allowing analysis of things like which team has a more balanced network.
This document contains exercises involving functions and graphs. There are multiple choice and free response questions testing understanding of properties of functions including domain, range, intercepts, maxima/minima, and graphing functions. Solutions are provided for checking work.
This document contains exercises and solutions related to monotone sequences, convergent subsequences, and the Bolzano-Weierstrass theorem from an introduction to real analysis course. It includes 4 problems examining properties of specific sequences, showing whether they are bounded/monotone and finding their limits, as well as examples of an unbounded sequence with a convergent subsequence and sequences that diverge. The solutions provide detailed proofs of the properties of each sequence using induction and algebraic manipulations.
1. The document defines various types of graphs including simple graphs, multigraphs, pseudographs, directed graphs, and bipartite graphs. It discusses properties like vertices, edges, degrees, adjacency, and special graph structures like complete graphs, cycles, wheels, and n-cubes.
2. Key terms are defined, such as the degree of a vertex, adjacent vertices, and handshaking theorems for both undirected and directed graphs. Special graph structures including complete graphs, cycles, wheels, and n-cubes are also described.
3. Bipartite graphs are introduced as simple graphs whose vertices can be partitioned into two disjoint sets such that every edge connects a vertex in one set to a
1. Dokumen membahas penggunaan integral tentu untuk menghitung luas daerah yang dibatasi kurva dan volume benda putar
2. Ada beberapa metode untuk menghitung volume benda putar yaitu metode cakram, cincin, dan kulit tabung
3. Beberapa contoh soal diberikan beserta penyelesaiannya untuk mengilustrasikan konsep-konsep tersebut
Dokumen menjelaskan tentang kombinasi linier vektor. Kombinasi linier adalah penjumlahan vektor yang diperoleh dengan mengalikan suatu skalar dengan vektor. Rentang vektor adalah himpunan semua kombinasi linier dari vektor-vektor tersebut.
The document provides notes on group theory. It discusses the definition of groups and examples of groups such as (Z, +), (Q, ×), and Sn. Properties of groups like Lagrange's theorem and criteria for subgroups are also covered. The notes then discuss symmetry groups, defining isometries of R2 and showing that the set of isometries forms a group. Symmetry groups G(Π) of objects Π in R2 are introduced and shown to be subgroups. Specific examples of symmetry groups like those of triangles, squares, regular n-gons, and infinite strips are analyzed. Finally, the concept of group isomorphism is defined and examples are given to illustrate isomorphic groups.
Introduction of metric dimension of circular graphs is connected graph , The distance and diameter , Resolving sets and location number then Examples . Application in facility location problems . is has motivation (Applications in Chemistry and Networks systems). Definitions of Certain Regular Graphs. Main Results for three graphs (Prism , Antiprism and generalized Petersen graphs .
This document discusses edge-magic total labelings of graphs. Some key points:
- An edge-magic total labeling assigns integers to the vertices and edges of a graph such that the sum of each edge's label and the labels of its incident vertices is a constant value called the magic sum.
- Necessary conditions for a graph to have an edge-magic total labeling include distinct sums for edge labels and no repeated vertex label sums for non-adjacent vertices.
- Some elementary counting formulas are derived relating the vertex labels, degrees, magic sum, and number of edges and vertices.
- It is shown that if a graph has an even number of edges and the sum of its vertices and edges is
The document discusses different types of topological spaces, including Hausdorff and non-Hausdorff spaces. It defines separation axioms like T0, T1, T2, etc. and explains that metric spaces and the real number line are Hausdorff. Non-Hausdorff spaces are exemplified using an equivalence relation on the space [0,1]∪[2,3]. Regular, normal and compact Hausdorff spaces are also discussed.
The document discusses surface integrals. It defines a surface integral as integrating a density function w over a surface σ. The surface σ is defined by a function z=f(x,y) over a domain D. The surface is partitioned into small patches, and each patch's area is approximated. The total mass of the surface is calculated as the limit of Riemann sums as the partitions approach zero. An example calculates the mass of a density function over a spherical surface portion.
METRIC DIMENSION AND UNCERTAINTY OF TRAVERSING ROBOTS IN A NETWORKFransiskeran
Metric dimension in graph theory has many applications in the real world. It has been applied to the
optimization problems in complex networks, analyzing electrical networks; show the business relations,
robotics, control of production processes etc. This paper studies the metric dimension of graphs with
respect to contraction and its bijection between them. Also an algorithm to avoid the overlapping between
the robots in a network is introduced.
METRIC DIMENSION AND UNCERTAINTY OF TRAVERSING ROBOTS IN A NETWORKgraphhoc
Metric dimension in graph theory has many applications in the real world. It has been applied to the
optimization problems in complex networks, analyzing electrical networks; show the business relations,
robotics, control of production processes etc. This paper studies the metric dimension of graphs with
respect to contraction and its bijection between them. Also an algorithm to avoid the overlapping between
the robots in a network is introduced.
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in GraphsIJERA Editor
In this paper we study about x-geodominating set, geodetic set, geo-set, geo-number of a graph G. We study the
binary operation, link vectors and some required results to develop algorithms. First we design two algorithms
to check whether given set is an x-geodominating set and to find the minimum x-geodominating set of a graph.
Finally we present another two algorithms to check whether a given vertex is geo-vertex or not and to find the
geo-number of a graph.
Algorithmic Aspects of Vertex Geo-dominating Sets and Geonumber in GraphsIJERA Editor
In this paper we study about x-geodominating set, geodetic set, geo-set, geo-number of a graph G. We study the
binary operation, link vectors and some required results to develop algorithms. First we design two algorithms
to check whether given set is an x-geodominating set and to find the minimum x-geodominating set of a graph.
Finally we present another two algorithms to check whether a given vertex is geo-vertex or not and to find the
geo-number of a graph.
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.
An analysis between exact and approximate algorithms for the k-center proble...IJECEIAES
This research focuses on the k-center problem and its applications. Different methods for solving this problem are analyzed. The implementations of an exact algorithm and of an approximate algorithm are presented. The source code and the computation complexity of these algorithms are presented and analyzed. The multitasking mode of the operating system is taken into account considering the execution time of the algorithms. The results show that the approximate algorithm finds solutions that are not worse than two times optimal. In some case these solutions are very close to the optimal solutions, but this is true only for graphs with a smaller number of nodes. As the number of nodes in the graph increases (respectively the number of edges increases), the approximate solutions deviate from the optimal ones, but remain acceptable. These results give reason to conclude that for graphs with a small number of nodes the approximate algorithm finds comparable solutions with those founds by the exact algorithm.
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.
This document appears to be a dissertation submitted for a Master of Philosophy in Mathematics degree. It discusses proximity in triangulations and quadrangulations. Key concepts discussed include definitions of planar graphs, triangulations, quadrangulations, eccentricity and radius of vertices. Computational results are presented on extremal structures that minimize connectivity values. Applications to measuring crowd density and mobility using proximity graphs inferred from WiFi sensor data are also described.
LADDER AND SUBDIVISION OF LADDER GRAPHS WITH PENDANT EDGES ARE ODD GRACEFULFransiskeran
The ladder graph plays an important role in many applications as Electronics, Electrical and Wireless
communication areas. The aim of this work is to present a new class of odd graceful labeling for the ladder
graph. In particular, we show that the ladder graph Ln with m-pendant Ln mk1 is odd graceful. We also
show that the subdivision of ladder graph Ln with m-pendant S(Ln) mk1 is odd graceful. Finally, we
prove that all the subdivision of triangular snakes ( k snake ) with pendant edges
1
( ) k S snake mk are odd graceful.
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.
GREY LEVEL CO-OCCURRENCE MATRICES: GENERALISATION AND SOME NEW FEATURESijcseit
Grey Level Co-occurrence Matrices (GLCM) are one of the earliest techniques used for image texture
analysis. In this paper we defined a new feature called trace extracted from the GLCM and its implications
in texture analysis are discussed in the context of Content Based Image Retrieval (CBIR). The theoretical
extension of GLCM to n-dimensional gray scale images are also discussed. The results indicate that trace
features outperform Haralick features when applied to CBIR.
STATE SPACE GENERATION FRAMEWORK BASED ON BINARY DECISION DIAGRAM FOR DISTRIB...csandit
This paper proposes a new framework based on Binary Decision Diagrams (BDD) for the graph distribution problem in the context of explicit model checking. The BDD are yet used to represent the state space for a symbolic verification model checking. Thus, we took advantage of high compression ratio of BDD to encode not only the state space, but also the place where each state will be put. So, a fitness function that allows a good balance load of states over the nodes of an homogeneous network is used. Furthermore, a detailed explanation of how to
calculate the inter-site edges between different nodes based on the adapted data structure is presented
STATE SPACE GENERATION FRAMEWORK BASED ON BINARY DECISION DIAGRAM FOR DISTRIB...cscpconf
This paper proposes a new framework based on Binary Decision Diagrams (BDD) for the graph distribution problem in the context of explicit model checking. The BDD are yet used to represent the state space for a symbolic verification model checking. Thus, we took advantage of high compression ratio of BDD to encode not only the state space, but also the place where each state will be put. So, a fitness function that allows a good balance load of states over the nodes of an homogeneous network is used. Furthermore, a detailed explanation of how to calculate the inter-site edges between different nodes based on the adapted data structure is presented.
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.
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.
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.
ALEXANDER FRACTIONAL INTEGRAL FILTERING OF WAVELET COEFFICIENTS FOR IMAGE DEN...sipij
The present paper, proposes an efficient denoising algorithm which works well for images corrupted with
Gaussian and speckle noise. The denoising algorithm utilizes the alexander fractional integral filter which
works by the construction of fractional masks window computed using alexander polynomial. Prior to the
application of the designed filter, the corrupted image is decomposed using symlet wavelet from which only
the horizontal, vertical and diagonal components are denoised using the alexander integral filter.
Significant increase in the reconstruction quality was noticed when the approach was applied on the
wavelet decomposed image rather than applying it directly on the noisy image. Quantitatively the results
are evaluated using the peak signal to noise ratio (PSNR) which was 30.8059 on an average for images
corrupted with Gaussian noise and 36.52 for images corrupted with speckle noise, which clearly
outperforms the existing methods.
This document discusses various graph parameters related to distance between vertices in a graph, including diameter, radius, and average distance. It examines how these parameters are defined, how they relate to each other and other graph properties, and how they behave in different graph classes. It also discusses characterizations of graph classes based on distance properties and generalizations to concepts like Steiner trees.
Graphs can be represented using adjacency matrices or adjacency lists. Common graph operations include traversal algorithms like depth-first search (DFS) and breadth-first search (BFS). DFS traverses a graph in a depth-wise manner similar to pre-order tree traversal, while BFS traverses in a level-wise or breadth-first manner similar to level-order tree traversal. The document also discusses graph definitions, terminologies, representations, elementary graph operations, and traversal methods like DFS and BFS.
This document discusses computing canonical labelings of digraphs. It begins by reviewing key concepts like digraphs, adjacency matrices, and isomorphisms. It notes that while many algorithms exist for undirected graphs, computing canonical labelings of digraphs remains challenging. The document then presents several new theoretical concepts for digraph canonical labeling, including mix diffusion degree sequences. It proposes using these concepts to systematically compute canonical labelings and proves several theorems to guide the algorithm. It describes four algorithms for calculating the canonical labeling of a digraph and notes the algorithms have been preliminarily verified through software testing.
Similar to AN APPLICATION OF Gd -METRIC SPACES AND METRIC DIMENSION OF GRAPHS (20)
SELECTIVE WATCHDOG TECHNIQUE FOR INTRUSION DETECTION IN MOBILE AD-HOC NETWORKFransiskeran
Mobile ad-hoc networks(MANET) is the collection of mobile nodes which are self organizing and are
connected by wireless links where nodes which are not in the direct range communicate with each other
relying on the intermediate nodes. As a result of trusting other nodes in the route, a malicious node can
easily compromise the security of the network. A black-hole node is the malicious node which drops the
entire packet coming to it and always shows the fresh route to the destination, even if the route to
destination doesn't exist. This paper describes a scheme that will detect the intrusion in the network in the
presence of black-hole node and its performance is compared with the previous technique. This novel
technique helps to increase the network performance by reducing the overhead in the network.
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The Link Quality Indicator (LQI) and Residual Energy have a fundamental impact on the network
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AN APPLICATION OF Gd -METRIC SPACES AND METRIC DIMENSION OF GRAPHS
1. International Journal on Applications of Graph Theory in Wireless Ad hoc Networks and Sensor
Networks(GRAPH-HOC) Vol.7, No.1, March 2015
DOI:10.5121/jgraphoc.2015.7101 1
AN APPLICATION OF Gd-METRIC SPACES AND
METRIC DIMENSION OF GRAPHS
Ms. Manjusha R1
and Dr. Sunny Kuriakose A2
1
Research Scholar, Union Christian College, Aluva, Kerala, India - 683102
2
Dean, Federal Institute of Science and Technology, Angamaly, Kerala, India - 683577
Abstract
The idea of metric dimension in graph theory was introduced by P J Slater in [2]. It has been found
applications in optimization, navigation, network theory, image processing, pattern recognition etc.
Several other authors have studied metric dimension of various standard graphs. In this paper we
introduce a real valued function called generalized metric +→×× RXXXGd : where == )/( WvrX
( ){ })(/),(),...,,(),,( 21 GVvvvdvvdvvd k ∈ , denoted dG and is used to study metric dimension of graphs. It
has been proved that metric dimension of any connected finite simple graph remains constant if dG
numbers of pendant edges are added to the non-basis vertices.
Keywords
Resolving set, Basis, Metric dimension, Infinite Graphs, dG -metric.
1. Introduction
Graph theory has been used to study the various concepts of navigation in an arbitrary space. A
work place can be denoted as node in a graph, and edges denote the connections between places.
The problem of minimum machine (or Robots) to be placed at certain nodes to trace each and
every node exactly once is worth investigating. The problem can be explained using networks
where places are interconnected in which, a navigating agent moves from one node to another in
the network. The places or nodes of a network where we place the machines (robots) are called
‘landmarks’. The minimum number of machines required to locate each and every node of the
network is termed as “metric dimension” and the set of all minimum possible number of
landmarks constitute “metric basis”.
A discrete metric like generalized metric [14] is defined on the Cartesian product XXX ×× of a
nonempty set X into +
R is used to expand the concept of metric dimension of the graph. The
definition of a generalized metric space is given in 2.6. In this type of spaces a non-negative real
number is assigned to every triplet of elements. Several other studies relevant to metric spaces are
being extended to G-metric spaces. Different generalizations of the usual notion of a metric
space were proposed by several mathematicians such as G¨ahler [17, 18] (called 2-metric spaces)
and Dhage [15, 16] (called D-metric spaces) have pointed out that the results cited by G¨ahler are
independent, rather than generalizations, of the corresponding results in metric spaces. Moreover,
it was shown that Dhage’s notion of D-metric space is flawed by errors and most of the results
established by him and others are invalid. These facts are determined by Mustafa and Sims [14] to
introduce a new concept in the area, called G-metric space.
2. International Journal on Applications of Graph Theory in Wireless Ad hoc Networks and Sensor
Networks(GRAPH-HOC) Vol.7, No.1, March 2015
2
The concept of metric dimension was introduced by P J Slater in [2] and studied independently
by Harary and Melter in [3]. Applications of this navigation of robots in networks are discussed
in [4] and in chemistry, while applications to problems of pattern recognition and image
processing, some of which involve the use of hierarchical structures are given in [5]. Besides
Kuller et.al. provide a formula and a linear time algorithm for computing the metric dimension of
a tree in [1]. On the other hand Chartrand et.al. in [7] characterize the graph with metric
dimension 1, n -1 and n -2. See also in [8] the tight bound on the metric dimension of unicyclic
graphs. Shanmukha and Sooryanarayana [9,10] compute the parameters for wheels, graphs
constructed by joining wheels with paths, complete graphs etc. In 1960’s a natural definition of
the dimension of a graph stated by Paul Erdos and state some related problems and unsolved
problems in [11]. Some other application including coin weighing problems and combinatorial
search and optimization [12]. The metric dimension of the Cartesian products of graph has been
studied by Peters-Fransen and Oellermann [13].
The metric dimension of various classes of graphs is computed in [3, 4, 5, 9, 10]. In [4, 5] the
results of [3] are corrected and in [9, 10] the results of [5] are refined.
2. Preliminaries
The basic definitions and results required in subsequent section are given in this section.
2.1. Definition
A graph ),( EVG = is an ordered pair consisting of a nonempty set )(GVV = of elements called
vertices and a set )(GEE = of unordered pair of vertices called edges.
Two vertices )(, GVvu ∈ are said to be adjacent if there is an edge )(GEuv ∈ joining them. The
edge )(GEuv ∈ is also said to be incident to vertices vandu . The degree of a vertex v , denoted by
)deg(v is the number of vertices in )(GV adjacent to it.
An edge of a graph is said to be a pendant edge if it is incident with only one vertex of the graph.
A uv -path is a sequence of distinct vertices vvvvu no == ,...,, 1 so that 1−iv is adjacent to iv for all
, 1i i n≤ ≤ , such a path is said to be of length n. A uu -path of length n is a cycle denoted by
nC .
A graph is said to be connected if there is a path between every two vertices. A complete graph is
a simple graph (a graph having no loops and parallel edges) in which each pair of distinct vertices
is joined by an edge.
2.2. Definition
A graph G is infinite if the vertex set )(GV is infinite. An infinite graph is locally finite if every
vertex has finite degree. An infinite graph is uniformly locally finite if there exists a positive
integer M such that the degree of each vertex is at most M . For example, the infinite path ∞P is
both locally finite and uniformly locally finite by taking 2=M .
2.3. Definition
If G is a connected graph, the distance ),( vud between two vertices V(G), ∈vu is the length of the
shortest path between them. Let { }kwwwW ,...,, 21= be an ordered set of vertices of G and let v be
3. International Journal on Applications of Graph Theory in Wireless Ad hoc Networks and Sensor
Networks(GRAPH-HOC) Vol.7, No.1, March 2015
3
a vertex of G. The representation )/( Wvr of v respect to W is the k-tuple
( )),(),...,,(),,( 21 kwvdwvdwvd . If distinct vertices of G have distinct representations (co-ordinates)
with respect toW , then W is called a resolving set or location set for G. A resolving set of
minimum cardinality is called a basis for G and this cardinality is called the metric dimension or
location number of G and is denoted by dim(G) or ( )Gβ .
For each landmark, the coordinate of a node ‘ v ’ in G having the elements equal to the cardinality
of the set W and th
i element of coordinate of ‘ v ’ equal to the length of the shortest path from the
th
i landmark to the vertex ‘ v ’ in G .
For example, consider the graph G of figure 1. The set },{ 211 vvW = is not a resolving set of G
Figure 1. Figure 2.
Since )/()1,1()/( 1413 WvrWvr == . Similarly, we can show that a set consisting of two distinct
vertices will not give distinct coordinates for the vertices inG . On the other hand, },,{ 3212 vvvW =
form a resolving set for G in figure 2, since the representation for the vertices in G with respect
to 2W are ( ) ( ) ( ) ( )1,0,1/,1,1,0/ 2221 == WvrWvr , ( ) ( )0,1,1/ 23 =Wvr , ( ) ( )1,1,1/ 24 =Wvr and it is the
minimum resolving set implying that dim( ) 3G = .
2.4. Remark
A graph can have more than one resolving set. For example consider the graph in figure 3. Here
we obtained two resolving sets namely {a,b} and {a,c}.
Figure 3. A graph with two resolving sets
4. International Journal on Applications of Graph Theory in Wireless Ad hoc Networks and Sensor
Networks(GRAPH-HOC) Vol.7, No.1, March 2015
4
2.5. Definition
Let X be a nonempty set. A dG - Metric or generalized metric is a function from XXX ×× into
+
R having the following properties:
XzyxzyxzyxGd ∈=== ,,forif0),,(
yxXyxyxxGd ≠∈≤ with,allfor),,(0
yzXzyxzyxGyxxG dd ≠∈≤ with,,allfor),,(),,(
)variablesthreetheallinsymmetry...(),,(),,(),,( === xzyGyzxGzyxG ddd &
)inequality(Rectangle,,,),,(),,(),,( XazyxzyaGaaxGzyxG ddd ∈∀+≤
2.6. Illustration
Let ),( dX be a metric space. Define +→×× RXXXGd : by
( )=zyxGd ,, ),(),(),( xzdzydyxd ++ is a dG -metric satisfying the above five conditions.
Conversely if ),( dGX is a dG -metric space, it is easy to verify that ),( dGdX is a metric space
where ( ) ( )),,(),,(
2
1
, yyxGyxxGyxd ddGd
+=
For,
a) ( ) ( ) 0),,(),,(
2
1
, ≥+= yyxGyxxGyxd ddGd
by (ii)
b) ( ) ( ) 0),,(),,(
2
1
, =+= xxxGxxxGxxd ddGd
by (i)
c) ( ) ( ) ( )xydxxyGxyyGyyxGyxxGyxd dd GddddG ,)),,(),,((
2
1
),,(),,(
2
1
, =+=+= by (iv)
d) ( ) ( )),,(),,(
2
1
, yyxGyxxGyxd ddGd
+=
[ ]),,(),,(),,(),,(
2
1
yyzGyzzGzzxGzxxG dddd +++≤
( ) ( )yzdzxd dd GG ,, +≤
Since ),,(),,(),,(),,( xxzGzzyGxxyGyxxG dddd +≤=
Similarly ),,(),,(),,( yyzGzzxGyyxG ddd +≤ by (v)
Now we recall a few results already published in [23]
2.7. Theorem [7]
The metric dimension of graph G is 1 if and only if G is a path.
Figure 4. (black colored vertices shows the metric basis for ∞P )
2.8. Theorem [7]
If nK is the complete graph with 1>n then 1)( −= nKnβ .
5. International Journal on Applications of Graph Theory in Wireless Ad hoc Networks and Sensor
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5
2.9. Theorem [1]
If nC is a cycle of length 2>n , then 2)( =nCβ .
2.9. Theorem [20]
If G is an infinite graph with finite metric dimension then it is uniformly locally finite.
The infinite graph ∞2P is uniformly locally finite with metric dimension equal to two.
Figure 4.
The converse of the above theorem is not true. That is a uniformly locally finite graph need not
have finite metric dimension. For example the infinite comp is uniformly locally finite but its
metric dimension is infinite.
Figure 5.
3. Main Results
3.1. Theorem
The metric dimension of the graph obtained by adding ‘n’ pendant edges to each of the ‘n’
vertices in the complete graph nK , 2>n is same as that of nK .
Proof: We have ( ) 1−= nKnβ . Let { } { }in vvvvW ,...,, 21= for some nii ≤≤1, be a basis for nK .
Since every vertices nK are adjacent to each other the coordinate of (n-1) vertices ijv j ≠, in W
has (n-1) components at which th
j component takes the value ‘0’ and the other components are
1’s with respect to W . Now the vertex Wvi ∉ is adjacent to the vertices in W , its coordinate
vector also has (n-1) components and that will be (1,1,…,1).
Suppose nmmm ,...,, 21 are the pendant edges added correspondingly to the vertices nvvv ,...,, 21 such
that ( ) njuvm jjj ≤≤= 1,, . Let the graph obtained in this way is denoted by ∑+=
=
n
j
jn mKK
1
.
We know that the coordinate of jv is (1,1,…,0(j
th
place),1,…,1). So for some j , 1),( =jj uvd and
6. International Journal on Applications of Graph Theory in Wireless Ad hoc Networks and Sensor
Networks(GRAPH-HOC) Vol.7, No.1, March 2015
6
since every vertex ( )nKVv∈ are adjacent to jv , 2),( =juvd for all those vertices jvv ≠ . Hence
the coordinate of ju will be (2,2,…,1(j
th
place),2,…,2).
That is, the coordinate of 1u is (1,2,…,2), 2u is (2,1,…,2)…, nu is (2,2,…,1) respectively. Thus
the vertices in the graph K obtained by adding ‘n’ pendant edges to each of the vertices in nK has
distinct coordinates with respect to W . Therefore W itself is the basis for nK and hence
( ) 1−= nKβ .
3.2. Illustration
Consider 5K (Figure 6). Here five pendant edges 51),,( ≤≤= juvm jjj are added at each of the
vertices 54321 and,,,, vvvvv respectively and shown that ( ) 415
5
1
5 =−=
∑+=
=j
jmKK ββ .
Figure 6.
The following corollary is about infinite graph with constant metric dimension.
3.3. Corollary
The above theorem holds for an infinite graph obtained by adding pendant edges
( ) njuvm jjj ≤≤= 1,, successively at each ju . Thus there exist infinite graphs with finite metric
dimension.
The development of uniformly locally finite (ULF)[19] graphs is based on the adjacency operator
A acting on the space of bounded sequences defined on the vertices. It has several applications
in spectral theory. The following theorem gives a simple result on uniformly locally finite graph.
3.4. Theorem
The infinite graph ∑+=
∞
=1j
jn mKK mentioned in theorem 3.1 is uniformly locally finite graph with
finite metric dimension.
Proof: By theorem 3.1 ( ) 1)(
1
−=∑+=
∞
=
nmKK
j
jnββ where ( ) ∞≤≤= juvm jjj 1,, . Since every
vertex is adjacent to each other in nK , 1)( −= nvd for )( nKVv∈ and the degree of the vertices ju
which is one of the end vertex in each of the edge added to nK is 2. Now fix a positive integer
1−= nM where 2>n . Then Mvd ≤)( for all Kv ∈ . Thus K is uniformly locally finite.
7. International Journal on Applications of Graph Theory in Wireless Ad hoc Networks and Sensor
Networks(GRAPH-HOC) Vol.7, No.1, March 2015
7
3.5. Theorem
Let G be connected graph with kG =)(β , { }kvvvW ,...,, 21= be the basis and
( ){ }.)(/)),(),...,,(,,()/( 21 GVvvvdvvdvvdWvrX k ∈== Define generalized metric or dG -metric
+→×× RXXXGd : by
( ) { }),(),,(),,(min,,
,,
zydzxdyxdzyxG k
RXzyx
d
rrrrrrrrr
rrr
⊆∈
=
Where d is the 2-metric defined from +→× RXX by ∑ −=
=
k
i
ii yxyxd
1
),(
rr
.
If mzyxGd =),,(
rrr
then the metric dimension of the super graph G
~
obtained by adjoining at most m
pendant edges to the vertices Wv∉ is same as that of G with respect toW . That is ( ) ( )GG ββ =
~
.
Proof: Let { }kvvvvW ...,,, 321= be the basis forG . Then the coordinate space
{ })(/),(),...,,(),,()/( 21 GVvvvdvvdvvdWvr k ∈= . Since kG =)(β , the coordinate of each vertex in G
contains ‘k’ components and they are distinct.
Let ( ) mzyxGd =,, . Now we add m pendant edges are added to suitable vertices Wv∉ . Suppose
the first pendant edge 1e is added at Wv j ∉ and ( )1
,1 ej vve = . The coordinate of jv is
)),(),...,,(),,(( 21 kjjj vvdvvdvvd and it is distinct from the coordinate of other vertices in G .
Figure 7.
Thus the coordinate of 1ev will be )1),(,...,1),(,1),(( 21 +++ kjjj vvdvvdvvd with respect to W and is
different from all other coordinates of the vertices in G since )),(),...,,(),,(( 21 kjjj vvdvvdvvd is
distinct from )),(),...,,(),,(( 21 kiii vvdvvdvvd , jini ≠= ,,...,2,1 . Hence keG =+ )( 1β . If the second
pendant edge is added at 1ev say ( )21
,2 ee vve = , then by the same argument as in the case of 1ev , the
coordinate of 2ev will be )1),(,...,1),(,1),(( 111 21 +++ keee vvdvvdvvd and it is distinct from all other
coordinates )),(),...,,(),,(( 21 kjjj vvdvvdvvd for 1,,...,2,1 ejnj == . Then obviously the coordinate of
the new vertex is distinct from all other vertices since each component in the coordinate of 2ev is
increased by one. Thus keeG =++ )( 21β .
Suppose the second pendant edge 2e is added to jivi ≠, in 1eG + and ( )2
,2 ei vve = . Here also the
coordinate of 2ev will be )1),(,...,1),(,1),(( 21 +++ kiii vvdvvdvvd . Hence keeG =++ )( 21β .
Therefore the result is true for 2,1=m . Assume that keeeG m =++++ − )....( 121β where
( )lejl vve ,= for 1,...,2,1, −=∉ mlWv j . If me is added at any lev then each of the ‘k’ components in
the coordinate of the vertex mev is increased by one and hence it is distinct from other coordinates.
If me is added to any vertex v in G not in W and not the end vertex of any of le , 1,...,2,1 −= ml ,
then the coordinate of mev will be )1),(,......1),(,1),(( 21 +++ kvvdvvdvvd and distinct from all other
coordinates of the vertices in the super graph 121 .... −++++ meeeG . Thus
8. International Journal on Applications of Graph Theory in Wireless Ad hoc Networks and Sensor
Networks(GRAPH-HOC) Vol.7, No.1, March 2015
8
keeeeG mm =+++++ − )....( 121β . Hence the result is true for m . Thus the theorem is true for any
integral value of +∈RzyxGd ),,(
rrr
.
3.6 Example
Consider a 5- vertex Kite say H (Figure 8). There ( ) ( )( ) ( ) ( ){ }2,3,1,2,1,10,1,1,0=X and
( ) { } XzyxzxdzydyxdMinzyxGd ∈= ,,,),(),,(),,(,,
r
∑ −==
=
2
1
),(where,2
i
ii yxyxd
Thus the minimum number of pendant edges that added to the Kite is 2. If these edges are added
to those vertices which are not in W namely 43 and vv with ( ) 221 =++ eeHβ .
Figure 8.
3.7. Example
Consider 4C
( ) 24 =Cβ with respect to { }21,vvW = (Figure 9). Then ( ) ( ) ( ) ( ){ }1,2,2,1,0,1,1,0)/( == WvrX
Figure 9.
By the definition of +→×× RXXXGd : , we have
( ) { } XzyxzxdzydyxdMinzyxGd ∈= ,,,),(),,(),,(,,
r
∑ −==
=
2
1
),(where,1
i
ii yxyxd
So one pendant edge is added to 4C .Suppose the pendant edge is added at Wv ∈1 and ( )1
,11 evve = .
Then the coordinate of 1ev is (1,2) with respect to W , but that is similar to the coordinate of 3v
(Figure 10). Therefore ( ) WeC respect towith214 ≠+β .
9. International Journal on Applications of Graph Theory in Wireless Ad hoc Networks and Sensor
Networks(GRAPH-HOC) Vol.7, No.1, March 2015
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Figure 10.
Similarly if 1e is added to Wv ∈2 , the coordinate of 1ev will be (2,1) and that is similar to the
coordinate of 4v (Figure 11). Thus 1e must be added to any of 3v or 4v . It will give a distinct
representation for the coordinates of the vertices in 14 eC + (Figure 12). That is 1e must be added
to the vertices not in .W
Figure 11.
Figure 12.
Note: Since W is not unique, ( ) 214 =+ eCβ with respect to another resolving set { }2,1
vvW e=
and ( ) ( )( ) ( ) ( ){ }1,3,2,2,0,21,1,2,0)/( =Wvr (Figure 13).
Figure 13.
4. Conclusion
This paper gives a measure that can be used in navigation space where the number of robots
required to navigate a work place kept constant. Extension of navigation space will lead us to
10. International Journal on Applications of Graph Theory in Wireless Ad hoc Networks and Sensor
Networks(GRAPH-HOC) Vol.7, No.1, March 2015
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infinite graphs and its properties. With the help of dG -metric and its properties we established
general concepts and results.
References
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[4] I. Javaid, M. T. Rahim and K. Ali, “ Families of Regular Graphs with constant Metric Dimension,”
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