Abstract
Let the square of a graphG , denoted by 2 G has same vertex set as in G and every two vertices u and v are joined in 2 G if and
only if they are joined in G by a path of length one or two. A subset D of vertices of 2 G is a double dominating set if every
vertex in 2 G is dominated by at least two vertices of D . The minimum cardinality double dominating set of 2 G is the double
domination number, and is denoted by 2
d G . In this paper, many bounds on 2
d G
were obtained in terms of elements of
G . Also their relationship with other domination parameters were obtained.
Key words: Graph, Square graph, Double dominating set, Double domination number.
Subject Classification Number: AMS-05C69, 05C70.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
The document discusses restrained domination in graphs. It defines a restrained dominating set of a lict graph as a set where every vertex not in the set is adjacent to a vertex in the set as well as another vertex not in the set. It studies the exact values of the restrained domination number (the minimum cardinality of a restrained dominating set) for some standard graphs. Bounds on the restrained domination number are obtained in terms of the number of vertices and edges of the graph. Relationships between the restrained domination number and other graph parameters like diameter and domination number are also investigated.
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.
A Note on Non Split Locating Equitable Dominationijdmtaiir
Let G = (V,E) be a simple, undirected, finite
nontrivial graph. A non empty set DV of vertices in a graph
G is a dominating set if every vertex in V-D is adjacent to
some vertex in D. The domination number (G) of G is the
minimum cardinality of a dominating set. A dominating set D
is called a non split locating equitable dominating set if for
any two vertices u,wV-D, N(u)D N(w)D,
N(u)D=N(w)D and the induced sub graph V-D is
connected.The minimum cardinality of a non split locating
equitable dominating set is called the non split locating
equitable domination number of G and is denoted by nsle(G).
In this paper, bounds for nsle(G) and exact values for some
particular classes of graphs were found.
A total dominating set D of graph G = (V, E) is a total strong split dominating set if the induced subgraph < V-D > is totally disconnected with atleast two vertices. The total strong split domination number γtss(G) is the minimum cardinality of a total strong split dominating set. In this paper, we characterize total strong split dominating sets and obtain the exact values of γtss(G) for some graphs. Also some inequalities of γtss(G) are established.
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.
The document discusses a new family of generalized distributions called the Kumaraswamy distributions (Kw-G distributions). These distributions extend common distributions like the normal, Weibull, and gamma distributions by introducing additional shape parameters. The key properties of the Kw-G distributions are:
1) They are defined on the interval (0,1) and are derived by applying a quantile function to a parent continuous distribution G(x).
2) Important special cases are the Kw-normal, Kw-Weibull, and Kw-gamma distributions.
3) Moments and probability weighted moments of the Kw-G distributions can be written as infinite weighted sums of the moments of the parent G distribution.
This document introduces the concept of weak triple connected domination number (γwtc) of a graph. A subset S of vertices is a weak triple connected dominating set if S is a weak dominating set and the induced subgraph <S> is triple connected. The γwtc is defined as the minimum cardinality of such a set. Some standard graphs are used to illustrate the concept and determine this number. Bounds on γwtc are obtained for general graphs, and its relationship to other graph parameters are investigated. The paper aims to develop this new graph invariant and establish basic results about weak triple connected domination.
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
The document discusses restrained domination in graphs. It defines a restrained dominating set of a lict graph as a set where every vertex not in the set is adjacent to a vertex in the set as well as another vertex not in the set. It studies the exact values of the restrained domination number (the minimum cardinality of a restrained dominating set) for some standard graphs. Bounds on the restrained domination number are obtained in terms of the number of vertices and edges of the graph. Relationships between the restrained domination number and other graph parameters like diameter and domination number are also investigated.
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.
A Note on Non Split Locating Equitable Dominationijdmtaiir
Let G = (V,E) be a simple, undirected, finite
nontrivial graph. A non empty set DV of vertices in a graph
G is a dominating set if every vertex in V-D is adjacent to
some vertex in D. The domination number (G) of G is the
minimum cardinality of a dominating set. A dominating set D
is called a non split locating equitable dominating set if for
any two vertices u,wV-D, N(u)D N(w)D,
N(u)D=N(w)D and the induced sub graph V-D is
connected.The minimum cardinality of a non split locating
equitable dominating set is called the non split locating
equitable domination number of G and is denoted by nsle(G).
In this paper, bounds for nsle(G) and exact values for some
particular classes of graphs were found.
A total dominating set D of graph G = (V, E) is a total strong split dominating set if the induced subgraph < V-D > is totally disconnected with atleast two vertices. The total strong split domination number γtss(G) is the minimum cardinality of a total strong split dominating set. In this paper, we characterize total strong split dominating sets and obtain the exact values of γtss(G) for some graphs. Also some inequalities of γtss(G) are established.
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.
The document discusses a new family of generalized distributions called the Kumaraswamy distributions (Kw-G distributions). These distributions extend common distributions like the normal, Weibull, and gamma distributions by introducing additional shape parameters. The key properties of the Kw-G distributions are:
1) They are defined on the interval (0,1) and are derived by applying a quantile function to a parent continuous distribution G(x).
2) Important special cases are the Kw-normal, Kw-Weibull, and Kw-gamma distributions.
3) Moments and probability weighted moments of the Kw-G distributions can be written as infinite weighted sums of the moments of the parent G distribution.
This document introduces the concept of weak triple connected domination number (γwtc) of a graph. A subset S of vertices is a weak triple connected dominating set if S is a weak dominating set and the induced subgraph <S> is triple connected. The γwtc is defined as the minimum cardinality of such a set. Some standard graphs are used to illustrate the concept and determine this number. Bounds on γwtc are obtained for general graphs, and its relationship to other graph parameters are investigated. The paper aims to develop this new graph invariant and establish basic results about weak triple connected domination.
Let G be a simple graph with n vertices, and λ1, · · · , λn be the eigenvalues of its adjacent matrix. The Estrada index of G is a graph invariant, defined as EE = i e n i 1 ,, is proposed as a measure of branching in alkanes. In this paper, we obtain two candidates which have the fourth largest EE among trees with n vertices
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
International journal of applied sciences and innovation vol 2015 - no 2 - ...sophiabelthome
This document discusses domatic subdivision stable and just excellent graphs. It begins by introducing key concepts related to domination in graphs such as domatic number, dominating sets, and private neighborhood. It then defines domatic subdivision stable graphs as graphs where the domatic number is unchanged after any edge subdivision. Just excellent graphs are defined as graphs where every vertex is contained in a unique minimum dominating set. The document proves several results about the relationships between these graph properties, including that just excellent graphs are not domatic subdivision stable, and that the domatic number decreases to 2 after any edge addition between vertices in the same dominating set of a just excellent graph with domatic number 3.
A study on connectivity in graph theory june 18 123easwathymaths
This document provides an introduction to connectivity of graphs. It begins with definitions of terms like bridges, cut vertices, connectivity, and edge connectivity. It then presents several theorems about when edges are bridges and vertices are cut vertices. It proves properties of trees related to cut vertices. The document establishes relationships between vertex and edge connectivity. It introduces the concepts of k-connectivity and discusses properties of complete graphs and trees in relation to connectivity.
The document defines several concepts related to domination in fuzzy graphs, including inverse split domination, non-split domination, and their bounds. It presents definitions for terms like dominating set, split dominating set, inverse dominating set, and establishes bounds on various domination numbers. Several theorems are provided with proofs, including that the inverse split domination number of a fuzzy tree is equal to its domination number, and that the inverse non-split domination number of a fuzzy graph is at most its minimum degree minus one. Coedge split and non-split domination in fuzzy graphs are also defined and properties presented.
International Journal of Computational Engineering Research(IJCER) ijceronline
nternational Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Strong (Weak) Triple Connected Domination Number of a Fuzzy Graphijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Abstract
A Roman dominating function on a graph G is a function f :V 0,1,2 satisfying the condition that every vertex uV for
which f u 0 is adjacent to at least one vertex vV for which f (v) 2 . The weight of a Roman dominating function is the
value ( ) ( )
v V
f V f v
. The Roman domination number R G of G is the minimum weight of a Roman dominating function on
G . A Roman dominating function on G is connected Roman dominating function of G if either 1 2 V V or 2 V is connected.
The connected Roman domination number RC G of G is the minimum weight of a connected Roman dominating function onG .
In this paper we establish the upper bounds, lower bounds and some equality results for RC G .
Keywords: Domination number, Roman domination number and Connected Roman domination number.
Subject Classification number: 05C69, 05C70.
Equi independent equitable domination number of cycle and bistar related 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 Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The degree equitable connected cototal dominating graph 퐷푐 푒 (퐺) of a graph 퐺 = (푉, 퐸) is a graph with 푉 퐷푐 푒 퐺 = 푉(퐺) ∪ 퐹, where F is the set of all minimal degree equitable connected cototal dominating sets of G and with two vertices 푢, 푣 ∈ 푉(퐷푐 푒 (퐺)) are adjacent if 푢 = 푣 푎푛푑 푣 = 퐷푐 푒 is a minimal degree equitable connected dominating set of G containing u. In this paper we introduce this new graph valued function and obtained some results
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
1. The document introduces concepts of equitable domination in fuzzy graphs. It defines fuzzy dominating sets and fuzzy domination numbers.
2. An equitable dominating set in a fuzzy graph is defined such that the degree of any dominating vertex is never more than 1 greater than the degree of the dominated vertex. The minimum cardinality of an equitable dominating set is the equitable domination number.
3. Properties of equitable domination in fuzzy graphs are explored, including results showing the domination number and equitable domination number are equal for regular and bi-regular fuzzy graphs. The concept of equitable isolates is also introduced.
Exact Solutions of the Klein-Gordon Equation for the Q-Deformed Morse Potenti...ijrap
This document presents an analysis of solving the Klein-Gordon equation for the q-deformed Morse potential using the Nikiforov-Uvarov method. The eigenfunctions and eigenvalues of the Klein-Gordon equation are obtained. It is found that the eigenfunctions can be expressed in terms of Laguerre polynomials. The energy eigenvalues and normalized eigenfunctions obtained agree with previous studies that used algebraic approaches.
Reciprocity Law For Flat Conformal Metrics With Conical SingularitiesLukasz Obara
The document is a thesis that establishes an analogue of Weil's reciprocity law for flat conformal metrics with conical singularities on Riemann surfaces. It introduces the necessary mathematical background, including definitions of isothermal coordinates and metrics with conical singularities. The main result proves a relationship between three flat conformally equivalent metrics, each with different conical singularities.
IOSR Journal of Mathematics(IOSR-JM) is an open access 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.
It is a algorithm used to find a minimum cost spanning tree for connected weighted undirected graph.This algorithm first appeared in Proceedings of the American Mathematical Society in 1956, and was written by Joseph Kruskal.
On the 1-2-3-edge weighting and Vertex coloring of complete graphijcsa
The document describes a 1-2-3 edge weighting and vertex coloring algorithm for complete graphs (K3q).
The algorithm assigns weights of 1, 2, or 3 to each edge in K3q such that the weighted degree of each vertex (the sum of edge weights connected to the vertex) is distinct. This results in a proper vertex coloring of K3q. The algorithm is proved to work for all K3q graphs. Some open problems related to generalizing the weighting to all graphs are also presented.
Connected domination in block subdivision graphs of graphsAlexander Decker
This document discusses connected domination in block subdivision graphs. It begins by defining key terms such as domination, connected domination, block subdivision graphs, and connected domination number. It then presents three theorems that establish bounds on the connected domination number of block subdivision graphs in terms of the number of vertices and cut vertices of the original graph G.
The document summarizes research on Roman dominating functions and total Roman dominating functions of the corona product graph of a path with a star. It begins by defining corona product graphs and reviewing relevant concepts in graph theory domination. It then defines Roman dominating functions and minimal Roman dominating functions. The document proves that a specific function is a minimal Roman dominating function of the corona product graph. It also discusses how this function satisfies the conditions of being a Roman dominating function and is minimal.
This document discusses two management techniques: time series analysis and work sampling. It provides details on:
1. Time series analysis techniques including trend analysis, forecasting, moving averages, weighted moving averages, and exponential smoothing which are used to predict patterns in data over time.
2. Work sampling which measures the proportion of time workers spend on different activities and can be used to determine standard times for tasks. The document outlines the steps for conducting a work sampling study.
3. An example of a work sampling study conducted over 1500 minutes to develop standard times for a cargo loading operation.
Let G be a simple graph with n vertices, and λ1, · · · , λn be the eigenvalues of its adjacent matrix. The Estrada index of G is a graph invariant, defined as EE = i e n i 1 ,, is proposed as a measure of branching in alkanes. In this paper, we obtain two candidates which have the fourth largest EE among trees with n vertices
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
International journal of applied sciences and innovation vol 2015 - no 2 - ...sophiabelthome
This document discusses domatic subdivision stable and just excellent graphs. It begins by introducing key concepts related to domination in graphs such as domatic number, dominating sets, and private neighborhood. It then defines domatic subdivision stable graphs as graphs where the domatic number is unchanged after any edge subdivision. Just excellent graphs are defined as graphs where every vertex is contained in a unique minimum dominating set. The document proves several results about the relationships between these graph properties, including that just excellent graphs are not domatic subdivision stable, and that the domatic number decreases to 2 after any edge addition between vertices in the same dominating set of a just excellent graph with domatic number 3.
A study on connectivity in graph theory june 18 123easwathymaths
This document provides an introduction to connectivity of graphs. It begins with definitions of terms like bridges, cut vertices, connectivity, and edge connectivity. It then presents several theorems about when edges are bridges and vertices are cut vertices. It proves properties of trees related to cut vertices. The document establishes relationships between vertex and edge connectivity. It introduces the concepts of k-connectivity and discusses properties of complete graphs and trees in relation to connectivity.
The document defines several concepts related to domination in fuzzy graphs, including inverse split domination, non-split domination, and their bounds. It presents definitions for terms like dominating set, split dominating set, inverse dominating set, and establishes bounds on various domination numbers. Several theorems are provided with proofs, including that the inverse split domination number of a fuzzy tree is equal to its domination number, and that the inverse non-split domination number of a fuzzy graph is at most its minimum degree minus one. Coedge split and non-split domination in fuzzy graphs are also defined and properties presented.
International Journal of Computational Engineering Research(IJCER) ijceronline
nternational Journal of Computational Engineering Research (IJCER) is dedicated to protecting personal information and will make every reasonable effort to handle collected information appropriately. All information collected, as well as related requests, will be handled as carefully and efficiently as possible in accordance with IJCER standards for integrity and objectivity.
Strong (Weak) Triple Connected Domination Number of a Fuzzy Graphijceronline
International Journal of Computational Engineering Research(IJCER) is an intentional online Journal in English monthly publishing journal. This Journal publish original research work that contributes significantly to further the scientific knowledge in engineering and Technology.
Abstract
A Roman dominating function on a graph G is a function f :V 0,1,2 satisfying the condition that every vertex uV for
which f u 0 is adjacent to at least one vertex vV for which f (v) 2 . The weight of a Roman dominating function is the
value ( ) ( )
v V
f V f v
. The Roman domination number R G of G is the minimum weight of a Roman dominating function on
G . A Roman dominating function on G is connected Roman dominating function of G if either 1 2 V V or 2 V is connected.
The connected Roman domination number RC G of G is the minimum weight of a connected Roman dominating function onG .
In this paper we establish the upper bounds, lower bounds and some equality results for RC G .
Keywords: Domination number, Roman domination number and Connected Roman domination number.
Subject Classification number: 05C69, 05C70.
Equi independent equitable domination number of cycle and bistar related 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 Journal of Engineering and Science Invention (IJESI) is an international journal intended for professionals and researchers in all fields of computer science and electronics. IJESI publishes research articles and reviews within the whole field Engineering Science and Technology, new teaching methods, assessment, validation and the impact of new technologies and it will continue to provide information on the latest trends and developments in this ever-expanding subject. The publications of papers are selected through double peer reviewed to ensure originality, relevance, and readability. The articles published in our journal can be accessed online.
The degree equitable connected cototal dominating graph 퐷푐 푒 (퐺) of a graph 퐺 = (푉, 퐸) is a graph with 푉 퐷푐 푒 퐺 = 푉(퐺) ∪ 퐹, where F is the set of all minimal degree equitable connected cototal dominating sets of G and with two vertices 푢, 푣 ∈ 푉(퐷푐 푒 (퐺)) are adjacent if 푢 = 푣 푎푛푑 푣 = 퐷푐 푒 is a minimal degree equitable connected dominating set of G containing u. In this paper we introduce this new graph valued function and obtained some results
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
1. The document introduces concepts of equitable domination in fuzzy graphs. It defines fuzzy dominating sets and fuzzy domination numbers.
2. An equitable dominating set in a fuzzy graph is defined such that the degree of any dominating vertex is never more than 1 greater than the degree of the dominated vertex. The minimum cardinality of an equitable dominating set is the equitable domination number.
3. Properties of equitable domination in fuzzy graphs are explored, including results showing the domination number and equitable domination number are equal for regular and bi-regular fuzzy graphs. The concept of equitable isolates is also introduced.
Exact Solutions of the Klein-Gordon Equation for the Q-Deformed Morse Potenti...ijrap
This document presents an analysis of solving the Klein-Gordon equation for the q-deformed Morse potential using the Nikiforov-Uvarov method. The eigenfunctions and eigenvalues of the Klein-Gordon equation are obtained. It is found that the eigenfunctions can be expressed in terms of Laguerre polynomials. The energy eigenvalues and normalized eigenfunctions obtained agree with previous studies that used algebraic approaches.
Reciprocity Law For Flat Conformal Metrics With Conical SingularitiesLukasz Obara
The document is a thesis that establishes an analogue of Weil's reciprocity law for flat conformal metrics with conical singularities on Riemann surfaces. It introduces the necessary mathematical background, including definitions of isothermal coordinates and metrics with conical singularities. The main result proves a relationship between three flat conformally equivalent metrics, each with different conical singularities.
IOSR Journal of Mathematics(IOSR-JM) is an open access 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.
It is a algorithm used to find a minimum cost spanning tree for connected weighted undirected graph.This algorithm first appeared in Proceedings of the American Mathematical Society in 1956, and was written by Joseph Kruskal.
On the 1-2-3-edge weighting and Vertex coloring of complete graphijcsa
The document describes a 1-2-3 edge weighting and vertex coloring algorithm for complete graphs (K3q).
The algorithm assigns weights of 1, 2, or 3 to each edge in K3q such that the weighted degree of each vertex (the sum of edge weights connected to the vertex) is distinct. This results in a proper vertex coloring of K3q. The algorithm is proved to work for all K3q graphs. Some open problems related to generalizing the weighting to all graphs are also presented.
Connected domination in block subdivision graphs of graphsAlexander Decker
This document discusses connected domination in block subdivision graphs. It begins by defining key terms such as domination, connected domination, block subdivision graphs, and connected domination number. It then presents three theorems that establish bounds on the connected domination number of block subdivision graphs in terms of the number of vertices and cut vertices of the original graph G.
The document summarizes research on Roman dominating functions and total Roman dominating functions of the corona product graph of a path with a star. It begins by defining corona product graphs and reviewing relevant concepts in graph theory domination. It then defines Roman dominating functions and minimal Roman dominating functions. The document proves that a specific function is a minimal Roman dominating function of the corona product graph. It also discusses how this function satisfies the conditions of being a Roman dominating function and is minimal.
This document discusses two management techniques: time series analysis and work sampling. It provides details on:
1. Time series analysis techniques including trend analysis, forecasting, moving averages, weighted moving averages, and exponential smoothing which are used to predict patterns in data over time.
2. Work sampling which measures the proportion of time workers spend on different activities and can be used to determine standard times for tasks. The document outlines the steps for conducting a work sampling study.
3. An example of a work sampling study conducted over 1500 minutes to develop standard times for a cargo loading operation.
Ebin Ephrem Elavathingal is an entrepreneur, researcher, and business strategist with experience founding startups and advising technology companies. He has co-founded organizations in the fields of education, media, and wireless sensor networks research. Ebin has expertise in various areas including business skills, engineering skills, programming skills, designing skills, and web development. He has applied for several patents, published papers in academic journals, and speaks regularly on topics related to entrepreneurship and technology.
2016-06 France Car Sales Mercedes June 2016Uli Kaiser
The document reports on Mercedes vehicle sales in France for June 2016 by model, ranking the top models from 1 to 23. It shows that Mercedes sold a total of 27,687 units in France that month. The top 5 models were the Classe A, Classe C, Classe B, GLA and GLC.
Esta presentacion habla de los grandes ídolos wen mi vida, elaborado por Anarely Dominguez Dominguez del grupo DN11 de la Universidad Tecnologica de Tulancingo Sede Cuautepec.
Justin Weaver is a mechanical engineering drafter and designer with 7 years of experience seeking a new full-time position. He has a Bachelor's degree in Technology Management and an Associate's degree in Computer Aided Drafting Technology. He is proficient in AutoCAD, Inventor, Solidworks, and other design software. His experience includes positions at Mauell Corporation, The Ames Companies, and Metso Minerals Industries where he designed and drafted mechanical parts and systems.
India is a highly diverse country with many languages, religions, ethnic groups, and castes coexisting. It has a rich cultural heritage spanning architecture, dance, music, art, and spiritual traditions. Music and dance are deeply ingrained in daily life from birth to marriage to death. Sports extend beyond cricket to include field hockey, kabaddi, chess, and others. India has contributed greatly to fields like medicine, architecture, and spirituality through practices like Ayurveda, yoga, and meditation. Its unity lies in this diversity, with peoples and cultures blending together harmoniously across its states.
Este documento descreve a casa espírita como um templo da vida cuja proposta é libertar as consciências e formar trabalhadores responsáveis em parceria com os Imortais para estabelecer o reino do bem nos corações. A casa espírita representa os esforços do Alto para implantar um oásis de paz na Terra através do trabalho contínuo de libertação dos tabus. O objetivo final é fornecer inspiração aos homens para a realização da paz.
Safe Harbor Statement: In conjunction with the provisions of the safe harbor section of the private securities litigation reform act of 1995, this document contains certain forward-looking statements pertaining to future anticipated projected plans, performance and developments, as well as other statements relating to future operations and results.
Any statements in this document that are not statements of historical fact may be considered to be forward-looking statements. Written words such as “may,” “will,” “expect,” “believe,” “anticipate,” “estimate,” “intends,” “goal,” objective,” “seek,” ”attempt” or variations of these or similar words, identify forward-looking statements.
These statements by their nature are estimates of future results only and involve substantial risks and uncertainties. There can be no assurance that actual results will not differ materially from expectations.
Este documento explora el uso de mundos virtuales como una nueva estrategia para enseñar idiomas extranjeros. Propone simular un escenario virtual como Machu Picchu para que los estudiantes aprendan mediante tareas colaborativas en las que usen su propio avatar y la lengua extranjera. Los autores concluyen que queda por responder preguntas sobre la implementación de metaversos con adolescentes y la reacción de los padres a este nuevo método de aprendizaje.
El ecosistema de financiación empresarial asinca vfinalRED CIDE Cide
Este documento discute la importancia de la financiación y el ecosistema de financiación para la innovación empresarial en Canarias. Explica que la financiación es clave para que las empresas sean más innovadoras y competitivas, aunque la financiación pública requiere más documentación y tiene tiempos más rígidos que la financiación privada. También destaca la necesidad de que las empresas innoven de manera continua y organizada para tener más éxito e incrementar sus ingresos.
Glioblastoma (GBM) is the most common and most malignant type of glioma. GBM typically appears as an enhancing mass on CT and MRI, and shows restricted diffusion on DWI. Advanced imaging techniques such as MR perfusion, MR spectroscopy, PET, and tractography provide additional information about glioblastomas. The standard treatment is maximal surgical resection followed by chemotherapy and radiation, but GBM often recurs due to its highly infiltrative nature. Advanced imaging helps distinguish GBM from other brain lesions and assess tumor grade and extent, aiding surgical planning and monitoring treatment response.
The document discusses student responses to a Bible correspondence course. It shares excerpts from letters written by students who have completed the course. The students praise the course for clearing up their misunderstandings and helping them better understand the Bible. They say the course exposed flaws in their previous religious teachings and helped them prove the accuracy of the scriptures cited in the lessons by looking them up directly. The students express enjoyment in studying the course and an interest in further biblical learning.
This document discusses methods of measuring soil moisture, including direct and indirect methods. Direct methods involve directly measuring the moisture content in soil samples through gravimetric, volumetric, or alcohol methods. Indirect methods measure water potential or tension, including gypsum blocks, tensiometers, neutron probes, and pressure plates. Gypsum blocks measure resistance which correlates to moisture, while tensiometers measure soil water tension. Neutron probes use radioactive materials to detect hydrogen atoms and calculate moisture content without disturbing soil. Indirect methods allow for continuous in-situ measurement compared to sampling with direct methods.
Definir concepto
Factores de riesgo vascular
Datos epidemiológicos
Evaluación clínica del paciente
Estudios de laboratorio
Etiología
Clasificación
Tratamiento abordaje actual
En las páginas siguientes se haran referencias de los puntos ya mencionados.
Automatic Water Sprinkling Through Real Time Estimation Of Soil Moisture Co...Rajesh Gangireddy
The project aims at reducing the wastage of water while irrigation and also at the same time provides a better solution for automation of irrigation systems.
The project uses a Soil moisture sensor (Hygrometer for soil) to detect the amount of moisture content in the soil. Once the moisture level reduces to a threshold level,the Micro-controller (we used Arduino MC Board) switches on the water pump until the soil is again moist. we also designed an Android Application - Jai Kisan V1.0 for manually controlling the time and the amount of water to be released to a plant. The project can be made useful for a large range for users - from a owner for a small garden in a metro to a multi acre farming farmer in a village.
This document provides definitions and theorems related to domination and strong domination of graphs. It begins with introductions to graph theory concepts like vertex degree. It then defines different types of domination like dominating sets, connected dominating sets, and k-dominating sets. Further definitions include total domination, strong domination, and dominating cycles. Theorems are provided that relate strong domination number to independence number and domination number. The document concludes by discussing applications of domination in fields like communication networks and distributing computer resources.
This document discusses non split locating equitable domination in graphs. It begins with definitions of terms like domination number and non split locating equitable dominating set. It then presents several theorems that establish bounds on the non split locating equitable domination number of a graph based on its properties. These include bounds related to the number of vertices, minimum degree, number of pendant vertices, and whether the graph is regular or a tree. The document also characterizes the graphs that achieve equality in some of the bounds. In general, it analyzes the non split locating equitable domination number and relates it to other graph parameters.
Trees amd properties slide for presentatonSHARANSASI1
The document discusses properties and parameters of graphs and trees. It defines key terms related to trees such as leaf, spanning tree, and properties like every tree having at least two leaves. It also defines graph parameters like diameter, radius, eccentricity, distance, center, and Wiener index. Formulas are given for counting trees and relations between graph parameters like maximum independent set size, matching size, vertex cover size, and edge cover size.
The document introduces the concept of a restrained triple connected dominating set and restrained triple connected domination number (γrtc) of a graph. A restrained triple connected dominating set is a restrained dominating set where the induced subgraph is triple connected. γrtc is defined as the minimum cardinality of a restrained triple connected dominating set. Bounds on γrtc are provided for general graphs. Exact values of γrtc are given for certain standard graphs like cycles, complete graphs, and complete bipartite graphs. Properties of γrtc are explored, such as the relationship between a graph and its spanning subgraphs.
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.
Abstract: An edge dominating set D of a fuzzy graph G= (σ, µ) is a non-split edge dominating set if the induced fuzzy sub graph H= (<e-d>, σ¢, µ¢) is connected. The split edge domination number γ¢ns(G)or γ¢ns is the minimum fuzzy cardinality of a non-split edge dominating set. In this paper we study a non-split edge dominating set of fuzzy graphs and investigate the relationship of γ¢ns(G)with other known parameter of G. Keywords: Fuzzy graphs, fuzzy domination, fuzzy edge domination, fuzzy non split edge domination number.
Title: Non Split Edge Domination in Fuzzy Graphs
Author: C.Y. Ponnappan, S. Basheer Ahamed, P. Surulinathan
ISSN 2350-1022
International Journal of Recent Research in Mathematics Computer Science and Information Technology
Paper Publications
The document discusses connectivity in graphs. It defines edge connectivity and vertex connectivity as numerical parameters that measure how connected a graph is. Edge connectivity is the minimum number of edges that need to be removed to disconnect the graph. Vertex connectivity is defined similarly for vertices. It provides examples and discusses properties like cut sets, bridges, and the relationship between these concepts and connectivity values. Menger's theorem relating the size of the minimum cut to the maximum number of disjoint paths is also covered.
The n -path graph PG (G) n of a graph G is a graph having the same vertex set as G and 2 vertices u and v in PG (G) n are adjacent if and only if there exist a path of length n between
u and v in G . In this paper we find n -path graph of some standard graphs. Bounds are given for the degree of a vertex in PG (G) n . We further characterise graphs G with PG2 (G) = G , PG2 (G) = G and PG G = Kn
The document discusses studies on petal graphs. It begins with Angel Sophia submitting her dissertation to Manonmaniam Sundaranar University for her Master's degree in Mathematics. The dissertation is about studies on petal graphs under the guidance of her advisor Dr. G. Easwaraprasad. It includes an acknowledgement, contents, and introduction chapters to set up the topic of studying the properties and behaviors of petal graphs.
The document discusses studies on petal graphs. It begins with Angel Sophia submitting her dissertation to Manonmaniam Sundaranar University for her Master's degree in Mathematics. The dissertation is about studies on petal graphs under the guidance of her advisor Dr. G. Easwaraprasad. It includes an introduction to petal graphs, basic definitions, and a discussion of the planarity of p-petal graphs with a main theorem.
The document describes results from graph theory, specifically matching theory, flows, coloring, and combinatorial nullstellensatz. It investigates classical theorems like Hall's theorem, Tutte's theorem, Dilworth's theorem, and Nash-Williams theorem on matchings in bipartite and general graphs. It also discusses network flows and Baranyai's theorem. The problems were taken from exercises in a book on modern graph theory.
This document discusses the concept of strong triple connected domination number (stc) of a graph. Some key points:
1. A subset S of vertices is a strong triple connected dominating set if S is a strong dominating set and the induced subgraph <S> is triple connected.
2. The strong triple connected domination number stc(G) is the minimum cardinality of a strong triple connected dominating set.
3. Some standard graphs for which the exact stc value is determined include paths, cycles, complete graphs, wheels, and more.
4. Bounds on stc(G) are established, such as 3 ≤ stc(G) ≤ p-1
IJCER (www.ijceronline.com) International Journal of computational Engineerin...ijceronline
This document defines and discusses the concept of paired triple connected domination number of a graph. It begins by reviewing existing concepts like domination number, connected domination number, and triple connected domination number. It then introduces the new concept of a paired triple connected dominating set as a triple connected dominating set where the induced subgraph also has a perfect matching. The paired triple connected domination number is defined as the minimum cardinality of such a set. The document explores properties of this number and its relationship to other graph parameters. Examples are provided to illustrate the definitions.
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.
The i(G)-graph is defined as a graph whose vertex set correspond 1 to 1 with the i(G)-sets of G . Two i(G)- sets say
S1 and 2 S are adjacent in i(G) if there exists a vertex S1
v , and a vertex wS2 such that v is adjacent to w and = { } { } 1 2 S S w v or equivalently = { } { } S2 S1 v w . In this paper we obtain i(G)-graph of some special graphs.
Graph theory with algorithms and its applicationsSpringer
The document discusses subgraphs, paths, and connected graphs in graph theory. It defines key terms like subgraph, spanning subgraph, walk, trail, path, connected graph, disconnected graph, and components. It also covers operations on graphs like union, intersection, ring sum, deletion, and fusion. Graphs can be decomposed or have induced subgraphs. The document proves theorems about connected graphs and provides definitions for cycles.
Welcome to International Journal of Engineering Research and Development (IJERD)IJERD Editor
call for paper 2012, hard copy of journal, research paper publishing, where to publish research paper,
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal,
This document defines and discusses the complementary perfect triple connected domination number of a graph. It begins by introducing concepts like triple connected graphs and triple connected dominating sets. It then defines a complementary perfect triple connected dominating set as a triple connected dominating set where the induced subgraph on the remaining vertices has a perfect matching. The complementary perfect triple connected domination number is the minimum cardinality of such sets. The document determines this number for some standard graph classes and establishes bounds for general graphs, exploring relationships with other graph parameters.
This document contains the exam for the course Mathematics M.Sc. Semester II held on 29 April 2013. It has 4 questions. Question 1 contains 8 multiple choice questions. Question 2 contains 9 short answer questions related to differential geometry topics like curves on surfaces, fundamental forms, geodesics, etc. Question 3 has 2 longer proof-based questions, one about constructing a curve with a given curvature and the other characterizing curves with constant curvature and torsion. Question 4 asks to show that a local diffeomorphism is an isometry if and only if the first fundamental forms of the domain and range surfaces are the same.
Similar to Bounds on double domination in squares of graphs (19)
Mechanical properties of hybrid fiber reinforced concrete for pavementseSAT Journals
Abstract
The effect of addition of mono fibers and hybrid fibers on the mechanical properties of concrete mixture is studied in the present
investigation. Steel fibers of 1% and polypropylene fibers 0.036% were added individually to the concrete mixture as mono fibers and
then they were added together to form a hybrid fiber reinforced concrete. Mechanical properties such as compressive, split tensile and
flexural strength were determined. The results show that hybrid fibers improve the compressive strength marginally as compared to
mono fibers. Whereas, hybridization improves split tensile strength and flexural strength noticeably.
Keywords:-Hybridization, mono fibers, steel fiber, polypropylene fiber, Improvement in mechanical properties.
Material management in construction – a case studyeSAT Journals
Abstract
The objective of the present study is to understand about all the problems occurring in the company because of improper application
of material management. In construction project operation, often there is a project cost variance in terms of the material, equipments,
manpower, subcontractor, overhead cost, and general condition. Material is the main component in construction projects. Therefore,
if the material management is not properly managed it will create a project cost variance. Project cost can be controlled by taking
corrective actions towards the cost variance. Therefore a methodology is used to diagnose and evaluate the procurement process
involved in material management and launch a continuous improvement was developed and applied. A thorough study was carried
out along with study of cases, surveys and interviews to professionals involved in this area. As a result, a methodology for diagnosis
and improvement was proposed and tested in selected projects. The results obtained show that the main problem of procurement is
related to schedule delays and lack of specified quality for the project. To prevent this situation it is often necessary to dedicate
important resources like money, personnel, time, etc. To monitor and control the process. A great potential for improvement was
detected if state of the art technologies such as, electronic mail, electronic data interchange (EDI), and analysis were applied to the
procurement process. These helped to eliminate the root causes for many types of problems that were detected.
Managing drought short term strategies in semi arid regions a case studyeSAT Journals
Abstract
Drought management needs multidisciplinary action. Interdisciplinary efforts among the experts in various fields of the droughts
prone areas are helpful to achieve tangible and permanent solution for this recurring problem. The Gulbarga district having the total
area around 16, 240 sq.km, and accounts 8.45 per cent of the Karnataka state area. The district has been situated with latitude 17º 19'
60" North and longitude of 76 º 49' 60" east. The district is situated entirely on the Deccan plateau positioned at a height of 300 to
750 m above MSL. Sub-tropical, semi-arid type is one among the drought prone districts of Karnataka State. The drought
management is very important for a district like Gulbarga. In this paper various short term strategies are discussed to mitigate the
drought condition in the district.
Keywords: Drought, South-West monsoon, Semi-Arid, Rainfall, Strategies etc.
Life cycle cost analysis of overlay for an urban road in bangaloreeSAT Journals
Abstract
Pavements are subjected to severe condition of stresses and weathering effects from the day they are constructed and opened to traffic
mainly due to its fatigue behavior and environmental effects. Therefore, pavement rehabilitation is one of the most important
components of entire road systems. This paper highlights the design of concrete pavement with added mono fibers like polypropylene,
steel and hybrid fibres for a widened portion of existing concrete pavement and various overlay alternatives for an existing
bituminous pavement in an urban road in Bangalore. Along with this, Life cycle cost analyses at these sections are done by Net
Present Value (NPV) method to identify the most feasible option. The results show that though the initial cost of construction of
concrete overlay is high, over a period of time it prove to be better than the bituminous overlay considering the whole life cycle cost.
The economic analysis also indicates that, out of the three fibre options, hybrid reinforced concrete would be economical without
compromising the performance of the pavement.
Keywords: - Fatigue, Life cycle cost analysis, Net Present Value method, Overlay, Rehabilitation
Laboratory studies of dense bituminous mixes ii with reclaimed asphalt materialseSAT Journals
Abstract
The issue of growing demand on our nation’s roadways over that past couple of decades, decreasing budgetary funds, and the need to
provide a safe, efficient, and cost effective roadway system has led to a dramatic increase in the need to rehabilitate our existing
pavements and the issue of building sustainable road infrastructure in India. With these emergency of the mentioned needs and this
are today’s burning issue and has become the purpose of the study.
In the present study, the samples of existing bituminous layer materials were collected from NH-48(Devahalli to Hassan) site.The
mixtures were designed by Marshall Method as per Asphalt institute (MS-II) at 20% and 30% Reclaimed Asphalt Pavement (RAP).
RAP material was blended with virgin aggregate such that all specimens tested for the, Dense Bituminous Macadam-II (DBM-II)
gradation as per Ministry of Roads, Transport, and Highways (MoRT&H) and cost analysis were carried out to know the economics.
Laboratory results and analysis showed the use of recycled materials showed significant variability in Marshall Stability, and the
variability increased with the increase in RAP content. The saving can be realized from utilization of recycled materials as per the
methodology, the reduction in the total cost is 19%, 30%, comparing with the virgin mixes.
Keywords: Reclaimed Asphalt Pavement, Marshall Stability, MS-II, Dense Bituminous Macadam-II
Laboratory investigation of expansive soil stabilized with natural inorganic ...eSAT Journals
This document summarizes a study on stabilizing expansive black cotton soil with the natural inorganic stabilizer RBI-81. Laboratory tests were conducted to evaluate the effect of RBI-81 on the soil's engineering properties. The tests showed that with 2% RBI-81 and 28 days of curing, the unconfined compressive strength increased by around 250% and the CBR value improved by approximately 400% compared to the untreated soil. Overall, the study found that RBI-81 effectively improved the strength properties of the black cotton soil and its suitability as a soil stabilizer was supported.
Influence of reinforcement on the behavior of hollow concrete block masonry p...eSAT Journals
Abstract
Reinforced masonry was developed to exploit the strength potential of masonry and to solve its lack of tensile strength. Experimental
and analytical studies have been carried out to investigate the effect of reinforcement on the behavior of hollow concrete block
masonry prisms under compression and to predict ultimate failure compressive strength. In the numerical program, three dimensional
non-linear finite elements (FE) model based on the micro-modeling approach is developed for both unreinforced and reinforced
masonry prisms using ANSYS (14.5). The proposed FE model uses multi-linear stress-strain relationships to model the non-linear
behavior of hollow concrete block, mortar, and grout. Willam-Warnke’s five parameter failure theory has been adopted to model the
failure of masonry materials. The comparison of the numerical and experimental results indicates that the FE models can successfully
capture the highly nonlinear behavior of the physical specimens and accurately predict their strength and failure mechanisms.
Keywords: Structural masonry, Hollow concrete block prism, grout, Compression failure, Finite element method,
Numerical modeling.
Influence of compaction energy on soil stabilized with chemical stabilizereSAT Journals
This document summarizes a study on the influence of compaction energy on soil stabilized with a chemical stabilizer. Laboratory tests were conducted on locally available loamy soil treated with a patented polymer liquid stabilizer and compacted at four different energy levels. The study found that increasing the compaction effort increased the density of both untreated and treated soil, but the rate of increase was lower for stabilized soil. Treating the soil with the stabilizer improved its unconfined compressive strength and resilient modulus, and reduced accumulated plastic strain, with these properties further improved by higher compaction efforts. The stabilized soil exhibited strength and performance benefits compared to the untreated soil.
Geographical information system (gis) for water resources managementeSAT Journals
This document describes a hydrological framework developed in the form of a Hydrologic Information System (HIS) to meet the information needs of various government departments related to water management in a state. The HIS consists of a hydrological database coupled with tools for collecting and analyzing spatial and non-spatial water resources data. It also incorporates a hydrological model to indirectly assess water balance components over space and time. A web-based GIS portal was created to allow users to access and visualize the hydrological data, as well as outputs from the SWAT hydrological model. The framework is intended to facilitate integrated water resources planning and management across different administrative levels.
Forest type mapping of bidar forest division, karnataka using geoinformatics ...eSAT Journals
Abstract
The study demonstrate the potentiality of satellite remote sensing technique for the generation of baseline information on forest types
including tree plantation details in Bidar forest division, Karnataka covering an area of 5814.60Sq.Kms. The Total Area of Bidar
forest division is 5814Sq.Kms analysis of the satellite data in the study area reveals that about 84% of the total area is Covered by
crop land, 1.778% of the area is covered by dry deciduous forest, 1.38 % of mixed plantation, which is very threatening to the
environmental stability of the forest, future plantation site has been mapped. With the use of latest Geo-informatics technology proper
and exact condition of the trees can be observed and necessary precautions can be taken for future plantation works in an appropriate
manner
Keywords:-RS, GIS, GPS, Forest Type, Tree Plantation
Factors influencing compressive strength of geopolymer concreteeSAT Journals
Abstract
To study effects of several factors on the properties of fly ash based geopolymer concrete on the compressive strength and also the
cost comparison with the normal concrete. The test variables were molarities of sodium hydroxide(NaOH) 8M,14M and 16M, ratio of
NaOH to sodium silicate (Na2SiO3) 1, 1.5, 2 and 2.5, alkaline liquid to fly ash ratio 0.35 and 0.40 and replacement of water in
Na2SiO3 solution by 10%, 20% and 30% were used in the present study. The test results indicated that the highest compressive
strength 54 MPa was observed for 16M of NaOH, ratio of NaOH to Na2SiO3 2.5 and alkaline liquid to fly ash ratio of 0.35. Lowest
compressive strength of 27 MPa was observed for 8M of NaOH, ratio of NaOH to Na2SiO3 is 1 and alkaline liquid to fly ash ratio of
0.40. Alkaline liquid to fly ash ratio of 0.35, water replacement of 10% and 30% for 8 and 16 molarity of NaOH and has resulted in
compressive strength of 36 MPa and 20 MPa respectively. Superplasticiser dosage of 2 % by weight of fly ash has given higher
strength in all cases.
Keywords: compressive strength, alkaline liquid, fly ash
Experimental investigation on circular hollow steel columns in filled with li...eSAT Journals
Abstract
Composite Circular hollow Steel tubes with and without GFRP infill for three different grades of Light weight concrete are tested for
ultimate load capacity and axial shortening , under Cyclic loading. Steel tubes are compared for different lengths, cross sections and
thickness. Specimens were tested separately after adopting Taguchi’s L9 (Latin Squares) Orthogonal array in order to save the initial
experimental cost on number of specimens and experimental duration. Analysis was carried out using ANN (Artificial Neural
Network) technique with the assistance of Mini Tab- a statistical soft tool. Comparison for predicted, experimental & ANN output is
obtained from linear regression plots. From this research study, it can be concluded that *Cross sectional area of steel tube has most
significant effect on ultimate load carrying capacity, *as length of steel tube increased- load carrying capacity decreased & *ANN
modeling predicted acceptable results. Thus ANN tool can be utilized for predicting ultimate load carrying capacity for composite
columns.
Keywords: Light weight concrete, GFRP, Artificial Neural Network, Linear Regression, Back propagation, orthogonal
Array, Latin Squares
Experimental behavior of circular hsscfrc filled steel tubular columns under ...eSAT Journals
This document summarizes an experimental study that tested circular concrete-filled steel tube columns with varying parameters. 45 specimens were tested with different fiber percentages (0-2%), tube diameter-to-wall-thickness ratios (D/t from 15-25), and length-to-diameter (L/d) ratios (from 2.97-7.04). The results found that columns filled with fiber-reinforced concrete exhibited higher stiffness, equal ductility, and enhanced energy absorption compared to those filled with plain concrete. The load carrying capacity increased with fiber content up to 1.5% but not at 2.0%. The analytical predictions of failure load closely matched the experimental values.
Evaluation of punching shear in flat slabseSAT Journals
Abstract
Flat-slab construction has been widely used in construction today because of many advantages that it offers. The basic philosophy in
the design of flat slab is to consider only gravity forces; this method ignores the effect of punching shear due to unbalanced moments
at the slab column junction which is critical. An attempt has been made to generate generalized design sheets which accounts both
punching shear due to gravity loads and unbalanced moments for cases (a) interior column; (b) edge column (bending perpendicular
to shorter edge); (c) edge column (bending parallel to shorter edge); (d) corner column. These design sheets are prepared as per
codal provisions of IS 456-2000. These design sheets will be helpful in calculating the shear reinforcement to be provided at the
critical section which is ignored in many design offices. Apart from its usefulness in evaluating punching shear and the necessary
shear reinforcement, the design sheets developed will enable the designer to fix the depth of flat slab during the initial phase of the
design.
Keywords: Flat slabs, punching shear, unbalanced moment.
Evaluation of performance of intake tower dam for recent earthquake in indiaeSAT Journals
Abstract
Intake towers are typically tall, hollow, reinforced concrete structures and form entrance to reservoir outlet works. A parametric
study on dynamic behavior of circular cylindrical towers can be carried out to study the effect of depth of submergence, wall thickness
and slenderness ratio, and also effect on tower considering dynamic analysis for time history function of different soil condition and
by Goyal and Chopra accounting interaction effects of added hydrodynamic mass of surrounding and inside water in intake tower of
dam
Key words: Hydrodynamic mass, Depth of submergence, Reservoir, Time history analysis,
Evaluation of operational efficiency of urban road network using travel time ...eSAT Journals
This document evaluates the operational efficiency of an urban road network in Tiruchirappalli, India using travel time reliability measures. Traffic volume and travel times were collected using video data from 8-10 AM on various roads. Average travel times, 95th percentile travel times, and buffer time indexes were calculated to assess reliability. Non-motorized vehicles were found to most impact reliability on one road. A relationship between buffer time index and traffic volume was developed. Finally, a travel time model was created and validated based on length, speed, and volume.
Estimation of surface runoff in nallur amanikere watershed using scs cn methodeSAT Journals
Abstract
The development of watershed aims at productive utilization of all the available natural resources in the entire area extending from
ridge line to stream outlet. The per capita availability of land for cultivation has been decreasing over the years. Therefore, water and
the related land resources must be developed, utilized and managed in an integrated and comprehensive manner. Remote sensing and
GIS techniques are being increasingly used for planning, management and development of natural resources. The study area, Nallur
Amanikere watershed geographically lies between 110 38’ and 110 52’ N latitude and 760 30’ and 760 50’ E longitude with an area of
415.68 Sq. km. The thematic layers such as land use/land cover and soil maps were derived from remotely sensed data and overlayed
through ArcGIS software to assign the curve number on polygon wise. The daily rainfall data of six rain gauge stations in and around
the watershed (2001-2011) was used to estimate the daily runoff from the watershed using Soil Conservation Service - Curve Number
(SCS-CN) method. The runoff estimated from the SCS-CN model was then used to know the variation of runoff potential with different
land use/land cover and with different soil conditions.
Keywords: Watershed, Nallur watershed, Surface runoff, Rainfall-Runoff, SCS-CN, Remote Sensing, GIS.
Estimation of morphometric parameters and runoff using rs & gis techniqueseSAT Journals
This document summarizes a study that used remote sensing and GIS techniques to estimate morphometric parameters and runoff for the Yagachi catchment area in India over a 10-year period. Morphometric analysis was conducted to understand the hydrological response at the micro-watershed level. Daily runoff was estimated using the SCS curve number model. The results showed a positive correlation between rainfall and runoff. Land use/land cover changes between 2001-2010 were found to impact estimated runoff amounts. Remote sensing approaches provided an effective means to model runoff for this large, ungauged area.
Effect of variation of plastic hinge length on the results of non linear anal...eSAT Journals
Abstract The nonlinear Static procedure also well known as pushover analysis is method where in monotonically increasing loads are applied to the structure till the structure is unable to resist any further load. It is a popular tool for seismic performance evaluation of existing and new structures. In literature lot of research has been carried out on conventional pushover analysis and after knowing deficiency efforts have been made to improve it. But actual test results to verify the analytically obtained pushover results are rarely available. It has been found that some amount of variation is always expected to exist in seismic demand prediction of pushover analysis. Initial study is carried out by considering user defined hinge properties and default hinge length. Attempt is being made to assess the variation of pushover analysis results by considering user defined hinge properties and various hinge length formulations available in literature and results compared with experimentally obtained results based on test carried out on a G+2 storied RCC framed structure. For the present study two geometric models viz bare frame and rigid frame model is considered and it is found that the results of pushover analysis are very sensitive to geometric model and hinge length adopted. Keywords: Pushover analysis, Base shear, Displacement, hinge length, moment curvature analysis
Effect of use of recycled materials on indirect tensile strength of asphalt c...eSAT Journals
Abstract
Depletion of natural resources and aggregate quarries for the road construction is a serious problem to procure materials. Hence
recycling or reuse of material is beneficial. On emphasizing development in sustainable construction in the present era, recycling of
asphalt pavements is one of the effective and proven rehabilitation processes. For the laboratory investigations reclaimed asphalt
pavement (RAP) from NH-4 and crumb rubber modified binder (CRMB-55) was used. Foundry waste was used as a replacement to
conventional filler. Laboratory tests were conducted on asphalt concrete mixes with 30, 40, 50, and 60 percent replacement with RAP.
These test results were compared with conventional mixes and asphalt concrete mixes with complete binder extracted RAP
aggregates. Mix design was carried out by Marshall Method. The Marshall Tests indicated highest stability values for asphalt
concrete (AC) mixes with 60% RAP. The optimum binder content (OBC) decreased with increased in RAP in AC mixes. The Indirect
Tensile Strength (ITS) for AC mixes with RAP also was found to be higher when compared to conventional AC mixes at 300C.
Keywords: Reclaimed asphalt pavement, Foundry waste, Recycling, Marshall Stability, Indirect tensile strength.
Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
ACEP Magazine edition 4th launched on 05.06.2024Rahul
This document provides information about the third edition of the magazine "Sthapatya" published by the Association of Civil Engineers (Practicing) Aurangabad. It includes messages from current and past presidents of ACEP, memories and photos from past ACEP events, information on life time achievement awards given by ACEP, and a technical article on concrete maintenance, repairs and strengthening. The document highlights activities of ACEP and provides a technical educational article for members.
Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapte...University of Maribor
Slides from talk presenting:
Aleš Zamuda: Presentation of IEEE Slovenia CIS (Computational Intelligence Society) Chapter and Networking.
Presentation at IcETRAN 2024 session:
"Inter-Society Networking Panel GRSS/MTT-S/CIS
Panel Session: Promoting Connection and Cooperation"
IEEE Slovenia GRSS
IEEE Serbia and Montenegro MTT-S
IEEE Slovenia CIS
11TH INTERNATIONAL CONFERENCE ON ELECTRICAL, ELECTRONIC AND COMPUTING ENGINEERING
3-6 June 2024, Niš, Serbia
Harnessing WebAssembly for Real-time Stateless Streaming PipelinesChristina Lin
Traditionally, dealing with real-time data pipelines has involved significant overhead, even for straightforward tasks like data transformation or masking. However, in this talk, we’ll venture into the dynamic realm of WebAssembly (WASM) and discover how it can revolutionize the creation of stateless streaming pipelines within a Kafka (Redpanda) broker. These pipelines are adept at managing low-latency, high-data-volume scenarios.
A review on techniques and modelling methodologies used for checking electrom...nooriasukmaningtyas
The proper function of the integrated circuit (IC) in an inhibiting electromagnetic environment has always been a serious concern throughout the decades of revolution in the world of electronics, from disjunct devices to today’s integrated circuit technology, where billions of transistors are combined on a single chip. The automotive industry and smart vehicles in particular, are confronting design issues such as being prone to electromagnetic interference (EMI). Electronic control devices calculate incorrect outputs because of EMI and sensors give misleading values which can prove fatal in case of automotives. In this paper, the authors have non exhaustively tried to review research work concerned with the investigation of EMI in ICs and prediction of this EMI using various modelling methodologies and measurement setups.
A SYSTEMATIC RISK ASSESSMENT APPROACH FOR SECURING THE SMART IRRIGATION SYSTEMSIJNSA Journal
The smart irrigation system represents an innovative approach to optimize water usage in agricultural and landscaping practices. The integration of cutting-edge technologies, including sensors, actuators, and data analysis, empowers this system to provide accurate monitoring and control of irrigation processes by leveraging real-time environmental conditions. The main objective of a smart irrigation system is to optimize water efficiency, minimize expenses, and foster the adoption of sustainable water management methods. This paper conducts a systematic risk assessment by exploring the key components/assets and their functionalities in the smart irrigation system. The crucial role of sensors in gathering data on soil moisture, weather patterns, and plant well-being is emphasized in this system. These sensors enable intelligent decision-making in irrigation scheduling and water distribution, leading to enhanced water efficiency and sustainable water management practices. Actuators enable automated control of irrigation devices, ensuring precise and targeted water delivery to plants. Additionally, the paper addresses the potential threat and vulnerabilities associated with smart irrigation systems. It discusses limitations of the system, such as power constraints and computational capabilities, and calculates the potential security risks. The paper suggests possible risk treatment methods for effective secure system operation. In conclusion, the paper emphasizes the significant benefits of implementing smart irrigation systems, including improved water conservation, increased crop yield, and reduced environmental impact. Additionally, based on the security analysis conducted, the paper recommends the implementation of countermeasures and security approaches to address vulnerabilities and ensure the integrity and reliability of the system. By incorporating these measures, smart irrigation technology can revolutionize water management practices in agriculture, promoting sustainability, resource efficiency, and safeguarding against potential security threats.
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
The rivalry between prominent international actors for dominance over Central Asia's hydrocarbon
reserves and the ancient silk trade route, along with China's diplomatic endeavours in the area, has been
referred to as the "New Great Game." This research centres on the power struggle, considering
geopolitical, geostrategic, and geoeconomic variables. Topics including trade, political hegemony, oil
politics, and conventional and nontraditional security are all explored and explained by the researcher.
Using Mackinder's Heartland, Spykman Rimland, and Hegemonic Stability theories, examines China's role
in Central Asia. This study adheres to the empirical epistemological method and has taken care of
objectivity. This study analyze primary and secondary research documents critically to elaborate role of
china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
pipeline politics, and winning states, according to this study, thanks to important instruments like the
Shanghai Cooperation Organisation and the Belt and Road Economic Initiative. According to this study,
China is seeing significant success in commerce, pipeline politics, and gaining influence on other
governments. This success may be attributed to the effective utilisation of key tools such as the Shanghai
Cooperation Organisation and the Belt and Road Economic Initiative.
1. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 454
BOUNDS ON DOUBLE DOMINATION IN SQUARES OF GRAPHS
M. H. Muddebihal1
, Srinivasa G2
1
Professor, Department of Mathematics, Gulbarga University, Karnataka, India, mhmuddebihal@yahoo.co.in
2
Assistant Professor, Department of Mathematics, B. N. M. I. T , Karnataka, India, gsgraphtheory@yahoo.com
Abstract
Let the square of a graph G , denoted by 2
G has same vertex set as in G and every two vertices u and v are joined in 2
G if and
only if they are joined in G by a path of length one or two. A subset D of vertices of 2
G is a double dominating set if every
vertex in 2
G is dominated by at least two vertices of D . The minimum cardinality double dominating set of 2
G is the double
domination number, and is denoted by 2
d G . In this paper, many bounds on 2
d G were obtained in terms of elements of
G . Also their relationship with other domination parameters were obtained.
Key words: Graph, Square graph, Double dominating set, Double domination number.
Subject Classification Number: AMS-05C69, 05C70.
--------------------------------------------------------------------***----------------------------------------------------------------------
1. INTRODUCTION
In this paper, we follow the notations of [1]. All the graphs
considered here are simple, finite and connected. As usual
( )p V G and ( )q E G denote the number of vertices
and edges of G , respectively.
In general, we use X to denote the subgraph induced by
the set of vertices X and ( )N v and N v denote the open
and closed neighborhoods of a vertex v , respectively. The
notation 0 1G G is the minimum number of
vertices(edges) is a vertex(edge) cover of G .
Also 0 1G G is the minimum number of vertices
(edges) is a maximal independent set of vertex (edge) of G .
Let deg v is the degree of a vertex v and as usual
G G denote the minimum (maximum) degree of
G . A vertex of degree one is called an end vertex and its
neighbor is called a support vertex. Suppose a support
vertex v is adjacent to at least two end vertices then it is
called a strong support vertex. A vertex v is called cut
vertex if removing it from G increases the number of
components of G .
The distance between two vertices u and v is the length of
the shortest u v - path in G . The maximum distance
between any two vertices in G is called the diameter,
denoted by diam G .
The square of a graph G , denoted by 2
G has the same
vertex set as in G and the two vertices u and v are joined
in 2
G if and only if they are joined in G by a path of length
one or two (see [1], [2]).
We begin by recalling some standard definitions from
domination theory.
A set S V is said to be a double dominating set of G , if
every vertex of G is dominated by at least two vertices of
S . The double domination number of G is denoted by
( )d G and is the minimum cardinality of a double
dominating set of G . This concept was introduced by F.
Harary and T. W. Haynes [3].
A dominating set ( )S V G is a restrained dominating set
of G , if every vertex not in S is adjacent to a vertex in S
and to a vertex in V S . The restrained domination number
of G , denoted by re G is the minimum cardinality of a
restrained dominating set of G . This concept was
introduced by G. S. Domke et. al.,[4].
A dominating set ( )S V G is said to be connected
dominating set of G , if the subgraph S is not
disconnected. The minimum cardinality of vertices in such a
set is called the connected domination number of G and is
denoted by ( )c G [5].
A subset 2
D V G is said to be a dominating set of 2
G ,
if every vertex not in D is adjacent to some vertex in D .
The domination number of 2
G , denoted by 2
G , is the
2. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 455
minimum cardinality of a dominating set of 2
G .
Similarly, a dominating set D of 2
G is said to be total
dominating set of 2
G , if for every vertex 2
v V G , there
exists a vertex u D , u v , such that u is adjacent to v
or if the subgraph D has no isolated vertex. The total
domination number of 2
G , denoted by 2
( )t G is the
minimum cardinality of total dominating set of 2
G .
Domination parameters in squares of graphs were
introduced by M. H. Muddebihal et. al.,(see [6] and [7]).
Analogously, a subset 2
D V G is said to be double
dominating set of 2
G , if every vertex in 2
G is dominated
by at least two vertices of D . The double domination
number of 2
G , denoted by 2
d G , is the minimum
cardinality of a double dominating set of 2
G . In this paper,
many bounds on 2
d G were obtained in terms of
elements of G . Also its relationship with other different
domination parameters were expressed.
2. RESULTS
Theorem 2.1:
a. For any cycle pC , with 3p vertices,
2
2, 3.
1, 0(mod3)
3
, .
3
d p
for p
p
C for p
p
otherwise
b. For any complete graph pK , with 2p vertices,
2
2d pK .
c. For any star 1,nK , with 2n vertices,
2
1, 2d nK .
d. For any wheel pW , with 4p vertices,
2
2d pW .
e. For any complete bipartite graph 1 2,p pK , with
1 2p p p vertices,
,1 2
2
2p pd K .
Theorem 2.2: For any connected graph G with 3p
vertices, 2
1
2
d
p
G
.
Proof: For 2p , 2
2
d
p
G
. For 3p , we prove
the result by induction process. Suppose 3p V in G ,
then 2
2
d
p
G
. Assume that the result is true for any
graph with p -vertices. Let G be a graph with 1p
vertices. Then by induction hypothesis, it follows that
2 1
2
d
p
G
. Hence the result is true for all graphs
with 3p vertices by induction process.
Theorem 2.3: For any connected graph G with 3p
vertices, 2 2
d G G p . Equality holds if and only if
3 3,G C P .
Proof: Let 1 2, ,..., kS v v v be the minimal set of vertices
which covers all the vertices in 2
G . Clearly, S forms a
dominating set of 2
G . Further, if there exists a vertex set
2
1V G S V in 2
G . Then '
S V D , where '
1V V in
2
G , be the set of vertices such that 2
v V G , there
exists two vertices in '
S V D . Further, since every
vertex of 2
G are adjacent to at least two vertices of 2
G ,
clearly D forms a double dominating set of 2
G . Therefore,
it follows that D S p . Hence 2 2
d G G p .
Suppose, 3 3,G C P . Then either 2 S D or D S p ,
which gives a contradiction in both cases.
Suppose, 3 3,G C P . Then in this case, 2 2 1 2D S .
Clearly, 3D S p . Therefore, 2 2
d G G p .
Theorem 2.4: For any connected ,p q - graph G ,
2 2
2 2dG G .
Proof: Suppose 2
1 2, ,..., nS v v v V G be the minimal
set of vertices which covers all the vertices, such that
, 3dist u v for all ,u v S . Then S forms a minimal
dominating set of 2
G . Further, if for every 2
v V G ,
there exists at least two vertices ,u w S such that ,u v ,
( )N v and ( )N u belongs to 2
V G S . Then S itself is a
double dominating set of 2
G . Otherwise, there exists at
least one vertex x N S such that S x D forms a
double dominating set of 2
G . Since for any
3. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 456
graph G with 2p , 2
2d G . Therefore, it follows that
2
2
D
S
. Clearly, 2 2
2 2dG G .
Theorem 2.5: For any connected ,p q -graphG ,
with 3p vertices, 2
d G G p . Equality holds for
3 4 3 4 5 7, , , , ,C C P P P P .
Proof: Let 1 1 2, ,..., mF v v v be the set of all non end
vertices in G . Suppose 1 2 1, ,..., ,kS v v v F k m , be
the minimal set of vertices which are at distance three
covers all the vertices of G . Then S itself forms a minimal
-set of G . Otherwise, there exists at least one vertex
v N S such that S v forms a minimal dominating set
of G . Now in 2
G , since 2
V G V G , let
1 2, ,..., iI u u u be the set of all strong support vertices.
Suppose '
1D I F , where '
1 1F F I be the minimum
set of vertices which covers all the vertices in 2
G , such that
for every vertex 2
v V G , there exists at least two
vertices ,u w D where iv I and
' 2
1 [ ]j kv F v V G D has at least two neighbors
which are either iv or jv . Then D forms a minimal double
dominating set of 2
G . Therefore, it follows that
D S v p . Hence 2
d G G p .
Suppose 3 4 3 4, , ,G C C P P . Then in this case, 2D and
2S p . Clearly, it follows that D S p . Therefore,
2
d G G p .
Suppose 5 7,G P P . Then in this case,
2
p
D
and
2
p
S
. Clearly, it follows that
2 2
p p
D S
. Therefore, 2
d G G p .
Theorem 2.6: For any connected ,p q - graph G with
3p vertices, 2
2
d
diam G
G p
.
Proof: For 2p , 2
2
d
diam G
G p
. Hence
consider 3p . Suppose there exists two vertices
,u v V G , which constitutes the longest path in G . Then
,dist u v diam G . Since 2
V G V G , there exists a
vertex set 1 2, ,..., iD v v v such that for every vertex
, 1jv D j i , there exists at least one vertex
, 1kv D k i . Also every vertex in 2
G is adjacent to at
least two vertices of D in 2
G . Then D forms a minimal
double dominating set of 2
G . Since 2D and the
diametral path includes at least two vertices. It follows that,
2 2D p diam G . Clearly, 2
2
d
diam G
G p
.
Theorem 2.7: For any nontrivial tree with 3p vertices
and m cut vertices, then 2
1d T m , equality holds
if 1,nT K , 2n .
Proof: Let 1 2, ,..., iB v v v be the set of all cut vertices in
T with B m . Suppose 1 2, ,..., jA v v v , 1 j i be the
set of cut vertices which are at a distance two from the end
vertices of T and A B . Now in 2
T , all the end vertices
are adjacent with jv A and { } { }B A . Now in 2
T ,
since 2
V T V T , for every vertex 2
v V T , there
exists at least two vertices ,u v B D in 2
T . Further,
since D covers all the vertices in 2
T , D itself forms a
minimal double dominating set of 2
T . Since every tree
T contains at least one cut vertex, it follows that
1D m . Hence 2
1d T m .
Suppose T is isomorphic to a star 1,nK . Then in this case,
2D and 1m . Therefore, it follows that
2
1d T m .
Theorem 2.8: For any connected ( , )p q -graph G ,
2 2
d tG G G .
Proof: For 2p , the result follows immediately. Hence,
let 3p . Suppose 1 1 2, ,..., nV v v v V G be the set of
all vertices with deg 2, 1iv i n . Then there exists at
least one vertex 1v V of maximum degree G . Now in
2
G , since 2
V G V G , let 1 1 2, ,..., kD v v v 1V in
2
G . Suppose 1D covers all the vertices in 2
G and if the
subgraph 1D has no isolated vertex, then 1D itself is a
minimal total dominating set of 2
G . Otherwise, there exists
a set 1D H , where 2
1H V G D , forms a minimal
total dominating set of 2
G . Now let 1 2 1, ,..., jD v v v V
in 2
G be the minimal set of vertices, which covers all the
vertices in 2
G . Suppose 2
v V G , there exists at least
4. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 457
two vertices ,u w D which are adjacent to at least one
vertex of D and at least two vertices of 2
V G D . Then
D forms a d - set of 2
G . Otherwise D I , where
2
I V G D , forms a minimal double dominating set of
2
G . Clearly, it follows that 1D I D H G .
Therefore, 2 2
d tG G G .
Theorem 2.9: For any connected ( , )p q -graph G ,
2
dG G . Equality holds if and only if 2G
with 3diam G .
Proof: If 1 1 2, ,..., nV v v v V G be the set of vertices
with deg 2, 1iv i n . Then 1 2 1, ,..., kS v v v V
forms a minimal dominating set of G . Now without loss of
generality in 2
G , since 2
V G V G . If
2 1 2, ,..., kV v v v be the set of vertices with deg 2kv .
If 2 ( )V V G , then the vertices which are at a distance at
least two are adjacent to each vertex of 2V in 2
G . Hence
1 2S V D where 1S S forms a minimal double
dominating set of 2
G . If 2V , then 3S V D where
3 1V V forms a minimal double dominating set of 2
G .
Further, since every vertex in 2
G is adjacent to atleast two
vertices of D , it follows that S D . Hence,
2
dG G .
Suppose 2G with 3diam G . Then in this case
1diam G and hence, 1S . Clearly, S D .
Therefore 2
dG G , a contradiction.
Further, if 2G with 3diam G . Then in this case,
4diam G . Clearly, D S . Therefore,
2
dG G , again a contradiction.
Hence 2
dG G if and only if 2G with
3diam G .
Theorem 2.10: For any connected ( , )p q -graph G ,
2
0 1d G p G . Equality holds for pK .
Proof: Let 1 2, ,..., nA v v v V G , where
deg 2, 1iv i n , be the minimum set of vertices which
covers all the edges of G , such that 0A G . Now in
2
G since 2
V G V G , let 1 2, ,..., kD v v v A be the
set of vertices such that for every vertex 2
v V G , there
exists at least two vertices ,u w D in 2
G . Further, if
D covers all the vertices in 2
G , then D itself is a double
dominating set of G . Clearly, it follows that 1D p A
and hence 2
0 1d G p G .
Suppose pG K . Then in this case, 1A p and 2D .
Clearly, it follows that 1D p A and
hence 2
0 1d G p G .
Theorem 2.11: For any connected ( , )p q -graph G ,
2
d tG G .
Proof: Let 1 2, ,..., nK u u u V G be the set of vertices
such that i jN u N u , where 1 i n , 1 j n .
Suppose there exists a minimal set
1 1 2, ,..., kK u u u N K , such that the subgraph
1K K has no isolated vertex. Further, if 1K K covers
all the vertices in G , then 1K K forms a minimal total
dominating set of G . Since 2
V G V G , there exists a
vertex set 1 2 1, ,..., mD v v v K K in 2
G , which covers
all the vertices in 2
G and for every vertex 2
v V G , there
exists at least two vertices ,u w D . Clearly, D forms a
minimal double dominating set of 2
G . Therefore, it follows
that 1D K K . Hence 2
d tG G .
Theorem 2.12: For any connected ,p q -graph G ,
2
0 1d G G . Equality holds for pK .
Proof: For 2p , the result is obvious. Hence let 3p .
Suppose 1 2, ,..., mF u u u V G be the set of all vertices
with deg 1, 1iv i m . Then '
F F , where
' '
,F V G F F N F forms a maximal independent
set of vertices, such that '
0F F G . Since
2
V G V G , let 2
1 1 2, ,..., nD v v v V G F and
1D N F . Suppose 2
2 1D V G D such that
1 2D D D forms a minimal set of vertices which covers
all the vertices in 2
G . Further, if for every vertex
2
v V G , there exists at least two vertices ,u w D .
Then D forms a minimal double dominating set of 2
G .
Since every graph G contains at least one independent
vertex, it follows that '
1D F F . Therefore,
2
0 1d G G .
5. IJRET: International Journal of Research in Engineering and Technology eISSN: 2319-1163 | pISSN: 2321-7308
_______________________________________________________________________________________
Volume: 02 Issue: 09 | Sep-2013, Available @ http://www.ijret.org 458
Suppose pG K . Then in this case, G contains exactly one
independent vertex and by Theorem 2.1(b), it follows that
2
0 1d G G .
Theorem 2.13: For any non-trivial tree T ,
2
1d reT T .
Proof: Let 1 2, ,..., nF v v v V T be the set of vertices
with deg 1, , 1i iv v F i n . Suppose for every
vertex v V T F , there exists a vertex u F and also a
vertex x V T F . Then F itself is a restrained
dominating set of T . Otherwise, there exists at least one
vertex w V G F , such that '
D F w forms a
minimal restrained dominating set of T . Let
1 2, ,..., kD u u u V F in 2
T be the minimal set of
vertices which are chosen such that 2
v V T , there
exists at least two vertices ,y z D . Further, since
D covers all the vertices in 2
T , clearly D forms a minimal
double dominating set of 2
T . Therefore, it follows that
'
1D D due to the distance between vertices of T is
one. Hence 2
1d reT T .
Theorem 2.14: For any connected graph G ,
2
1d cG G .
Proof: Suppose 1 2, ,..., nC v v v V G be the set of all
cut vertices in G . Further, if C I , where I N C with
deg 2,i iv v I be the minimal set of vertices which
covers all the vertices in G and if the sub graph C I is
connected. Then C I forms a minimal connected
dominating set of G . Let 1 2, ,..., kD v v v be the minimal
set of vertices which covers all the vertices in 2
G . Suppose
for every vertex 2
v V G , there exists at least two
vertices ,u w D . Then D itself forms a minimal double
dominating set of 2
G . Therefore, it follows that
1D C I and hence 2
1d cG G .
REFERENCES
[1]. F. Harary, Graph Theory, Adison-Wesley, Reading,
Mass., 1972.
[2]. F. Harary and I. C. Ross, The square of a tree, Bell
System Tech. J. 39, 641-647, 1960.
[3]. F. Harary and T. W. Haynes, Double domination in
graphs, Ars Combinatorica 55, 201-213, 2000.
[4]. G. S. Domke, J. H. Hattingh, S. T. Hedetniemi, R. C.
Laskar and L. R. Markus, Restrained Domination in
Graphs, Discrete Mathematics 203, 61–69, 1999.
[5]. E. Sampathkumar and H. B. Walikar, The Connected
Domination Number of Graphs, J.Math.Phy.Sci. 13,
607-613, 1979.
[6]. M. H. Muddebihal, G. Srinivasa and A. R. Sedamkar,
Domination in Squares of Graphs, Ultra Scientist,
23(3)A, 795–800, 2011.
[7] M. H. Muddebihal and G. Srinivasa, Bounds on Total
Domination in Squares of Graphs, International Journal
of Advanced Computer and Mathematical Sciences,
4(1), 67–74, 2013