Intersection Graph Research Articles

Total Coloring of the Generalized Corona Product of Graphs

Total Coloring of the Generalized Corona Product of Graphs

DOMINATOR CHROMATIC NUMBER OF LEXICOGRAPHIC PRODUCT OF PATH WITH CERTAIN GRAPHS

A dominator coloring of a graph is a proper coloring in which every vertex is adjacent to at least one vertex from each color class. The minimum number of colors required to achieve a dominator coloring of is called the dominator chromatic number, denoted as . In this paper, we obtain the dominator chromatic number of the lexicographic product of a path with certain graphs.

Locally Identifying Coloring of Strong Product of Graphs

Locally Identifying Coloring of Strong Product of Graphs

Analysis of Approximation Methods of Laplacian Eigenvectors of the Kronecker Product of Graphs

This paper analyzes two approximation methods for the Laplacian eigenvectors of the Kronecker product, as recently presented in the literature. We enhance the approximations by comparing the correlation coefficients of the eigenvectors, which indicate how well an arbitrary vector approximates a matrix’s eigenvector. In the first method, some correlation coefficients are explicitly calculable, while others are not. In the second method, only certain coefficients can be estimated with good accuracy, as supported by empirical and theoretical evidence, with the rest remaining incalculable. The primary objective is to evaluate the accuracy of the approximation methods by analyzing and comparing limited sets of coefficients on one hand and the estimation on the other. Therefore, we compute the extreme values of the mentioned sets and theoretically compare them. Our observations indicate that, in most cases, the relationship between the majority of the values in the first set and those in the second set reflects the relationship between the remaining coefficients of both approximations. Moreover, it can be observed that each of the sets generally contains smaller values compared to the values found among the remaining correlation coefficients. Finally, we find that the performance of the two approximation methods is significantly influenced by imbalanced graph structures, exemplified by a class of almost regular graphs discussed in the paper.

Distance-Hereditary Properties in Product Graphs

The origin of distance-hereditary graphs can be traced back to as early as 1977. Edward Howorka first introduced that a connected graph [Formula: see text] is distance-hereditary if and only if [Formula: see text] for all [Formula: see text] for every connected induced subgraph [Formula: see text] of [Formula: see text] In this paper, we study the distance-hereditary properties of product graphs.

Unexpected automorphisms in direct product graphs

Unexpected automorphisms in direct product graphs

Hierarchical product graphs and their prime factorization

Hierarchical product graphs and their prime factorization

The minimum orientable genus of the repeated Cartesian product of graphs

Determining the minimum genus of a graph is a fundamental optimisation problem in the study of network design and implementation as it gives a measure of non-planarity of graphs. In this paper, we are concerned with determining the smallest value of g such that a given graph G has an embedding on the orientable surface of genus g. In particular, we consider the Cartesian product of graphs since this is a well studied graph operation which is often used for modeling interconnection networks. The s-cube Qi(s) is obtained by taking the repeated Cartesian product of i complete bipartite graphs Ks,s. We determine the genus of the Cartesian product of the 2r-cube with the repeated Cartesian product of cycles and of the Cartesian product of the 2r-cube with the repeated Cartesian product of paths.

Graph Entropy Based on Wiener Polarity Index Under Four Kinds of Graph Operations

The Wiener polarity index of a graph <i>G</i>, is the number of unordered pairs of vertices that are at distance 3 in <i>G</i>. This index can reflect the specific distance relation between vertices in the graph, and provides a new way for the study of graph structure. In this paper, the graph entropy based on Wiener polarity index defined. Based on the above definition of graph entropy, it compares the graph entropy of path and balanced double star graphs based on Wiener polarity index. The expressions of graph entropy based on Wiener polarity index for trees with diameter <i>d ≥ 3</i> are studied under four graph operations: tensor product, strong product, Cartesian product and composite graph.

Quantumlike Product States Constructed from Classical Networks.

Can complex classical systems be designed to exhibit superpositions of tensor products of basis states, thereby mimicking quantum states? We exhibit a one-to-one map between the product basis of quantum states comprising an arbitrary number of qubits and the eigenstates of a construction comprising classical oscillator networks. Specifically, we prove the existence of this map based on Cartesian products of graphs, where the graphs depict the layout of oscillator networks. We show how quantumlike gates can act on the classical networks to allow quantumlike operations in the state space.

Monoid algebras and graph products

Monoid algebras and graph products

Omega Indices of Strong and Lexicographic Products of Graphs.

The degree sequence of a graph is the list of its vertex degrees arranged in usually increasing order. Many properties of the graphs realized from a degree sequence can be deduced by means of a recently introduced graph invariant called omega invariant. We used the definitions of the considered graph products together with the list of degree sequences of these graph products for some well-know graph classes. Naturally, the vertex degree and edge degree partitions are used. As the main theme of the paper is the omega invariant, we frequently used the definition and fundamental properties of this very new invariant for our calculations. Also, some algebraic properties of these products are deduced in line with some recent publications following the same fashion. In this paper, we determine the degree sequences of strong and lexicographic products of two graphs and obtain the general form of the degree sequences of both products. We obtain a general formula for the omega invariant of strong and lexicographic products of two graphs. The algebraic structures of strong and lexicographic products are obtained. Moreover, we prove that strong and lexicographic products are not distributive over each other. We have obtained the general expression for degree sequences of two important products of graphs and a general expression for omega invariants of strong and lexicographic products. Furthermore, we have obtained algebraic structures of strong and lexicographic products in terms of their degree sequences. Also, it has been found that the disruptive property does not hold for strong and lexicographic products.

Acyclic coloring of two operations of graphs

Abstract The coloring theory of graphs is a very important direction in graph theory. The graph coloring problem has a strong application background. Many practical problems such as timetabling problems, frequency allocation problems, traffic arrangements, circuit design, and storage problems can be transformed into graph coloring problems. A substantial amount of scholarly work has been dedicated to the exploration of acyclic coloring, resulting in a plethora of significant findings that significantly contribute to the theoretical framework of vertex coloring in graphs. An acyclic coloring of a graph G refers to a proper vertex coloring where the subgraph resulting from any pair of color classes does not encompass any cycles. We analyze acyclic coloring in splitting graphs and Cartesian products of paths and cycles by using the method of structural coloring, reductive proof, and mathematical induction. Moreover, we present the precise acyclic chromatic numbers for splitting graphs of cycles and Cartesian products of paths and cycles.

Injective coloring of edge corona product of graphs

Abstract The injective chromatic number χi(G) of a graph G is the minimum number of colors needed to color the vertices of G such that different vertices with the same neighbor are colored differently. In this here, we get some exact values of the injective chromatic number of the edge corona product of some graphs.

Connectivity in Cartesian products of fuzzy graphs: application to decision making

Connectivity in Cartesian products of fuzzy graphs: application to decision making

Strength of Vertex Irregularity of Some Product of Graphs

“G is a connected graph with number of vertices . Let be the set of vertices and be the set of edges of . Let be a function which assigns the integers {1, 2, …n} not necessarily distinct to the edges of . The weight of a vertex denoted by is the sum of all the labels of the edges which are incident with . Then the function is called a vertex irregular labelling if all the weight of the vertices are distinct. The strength of vertex irregularity of the graph denoted by is the least positive integer n such that there exists a vertex irregular labelling . In this paper, we determine the strength of vertex irregularity of some edge corona product of graphs”. e determine the strength of vertex irregularity of some edge corona product of graphs”.

Advanced clique algorithms for protein product graphs

Advanced clique algorithms for protein product graphs

Fault-Tolerant Strong Metric Dimension of Rooted Product Graphs

Fault-Tolerant Strong Metric Dimension of Rooted Product Graphs

Scalable Estimation and Two-Sample Testing for Large Networks via Subsampling

In recent years, large networks are routinely used to represent data from many scientific fields. Statistical analysis of these networks, such as estimation and hypothesis testing, has received considerable attention. However, most of the methods proposed in the literature are computationally expensive for large networks. In this article, we propose a subsampling-based method to reduce the computational cost of estimation and two-sample hypothesis testing. The idea is to divide the network into smaller subgraphs with an overlap region, then draw inference based on each subgraph, and finally combine the results together. We first develop the subsampling method for random dot product graph models, and establish theoretical consistency of the proposed method. Then we extend the subsampling method to a more general setup and establish similar theoretical properties. We demonstrate the performance of our methods through simulation experiments and real data analysis. Supplemental materials for the article are available online. The code is available in the following GitHub repository: https://github.com/kchak19/SubsampleTestingNetwork. Supplementary materials for this article are available online.

Graph domination in rooted products: equitable and outdegree equitable approaches

In this paper, equitability related domination parameters such as equitable domination number and outdegree equitable domination number are investigated in the context of rooted product of graphs. Obtained a lower bound for the equitable domination number of rooted product of graphs. Obtained upper as well as lower bound for the outdegree equitable domination number of rooted product of graphs. Some particular rooted product of graphs are examined in detail.