Power Edge Domination Number of Certain Graphs in its Corona Product

Graph G is usually written by G = (V, E) is graph G where V(G) is vertex set on graph G and E(G) is edge set on graph G. Graph G used in this study is only on simple and undirected graphs. Dominating set (DS) is graph G which have a vertex set D, where each vertex in D can dominate the neighboring vertices, in other words every vertex from u ∈ V(G) − D is adjacent to verted v ∈ D. The minimum cardinality of dominating set is called by domination number, symbolized by γ(G). Locating dominating set (LDS) is dominating set with additional condition. A graph G = (V, E) is said to be locating dominating set if the set of vertex dominator D satisfies every vertex that is not D, that is V − D has a different intersection set with D. The minimum cardinality of locating dominating set is called by locating domination number, symbolized by γL (G). In this paper we will determine the LDS on edge corona product. The edge corona product of graph is development of corona product graph. The edge corona of two graphs G and H is obtained by taking one copy of G and |E(G)| copies of H and joining each end vertices of i-th edge of G to every vertex in the i-th copy of H, symbolized by G ⋄ H. The results in this study are shown that there is a relation between the locating dominating set on the basic graph and its operation.

A total coloring of a graph [Formula: see text] is an assignment of colors to the elements of the graph [Formula: see text] such that no adjacent vertices and edges receive the same color. The total chromatic number of a graph [Formula: see text], denoted by [Formula: see text], is the minimum number of colors that suffice in a total coloring. Behzad and Vizing conjectured that for any simple graph [Formula: see text], [Formula: see text], where [Formula: see text] is the maximum degree of [Formula: see text]. In this paper, we prove the tight bound of the total coloring conjecture for the three types of corona products (vertex, edge and neighborhood) of graphs.

A graph is equitably k-colorable if its vertices can be partitioned into k independent sets in such a way that the numbers of vertices in any two sets differ by at most one. The smallest k for which such a coloring exists is known as the equitable chromatic number of G and denoted by 𝜒=(G). It is known that the problem of computation of 𝜒=(G) is NP-hard in general and remains so for corona graphs. In this paper we consider the same model of coloring in the case of corona multiproducts of graphs. In particular, we obtain some results regarding the equitable chromatic number for the l-corona product G ◦l H, where G is an equitably 3- or 4-colorable graph and H is an r-partite graph, a cycle or a complete graph. Our proofs are mostly constructive in that they lead to polynomial algorithms for equitable coloring of such graph products provided that there is given an equitable coloring of G. Moreover, we confirm the Equitable Coloring Conjecture for corona products of such graphs. This paper extends the results from [H. Furmánczyk, K. Kaliraj, M. Kubale and V.J. Vivin, Equitable coloring of corona products of graphs, Adv. Appl. Discrete Math. 11 (2013) 103–120].

It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G1×G2 is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs).

The Rainbow Connection Number of an n-Crossed Prism Graph and its Corona Product with a Trivial Graph

In an edge-colored graph (where adjacent edges may have the same color), a rainbow path is a path whose edge colors are all distinct. The coloring is called a rainbow coloring if any two vertices can be connected by a rainbow path. The rainbow connection number rc ( G ) is the smallest number of colors in a rainbow coloring of G . The corona product G ∘ H of two graphs G and H is constructed from one copy of G and n = | V ( G ) | disjoint copies of H such that the i -th vertex of G is joined to all vertices in the i -th copy of H , for each i ∈{ 1 , … ,n } . Several resuls on the rainbow connection number of corona product have been published, but there are inaccuracies. In this paper, we close the gaps and add new results. The strong variant of rainbow connection number is also discussed.

The corona product of two graphs Gn1 and Gn2 is one of the most important graph operations that allows us to build a complex graph by means of simple graphs. In this paper, firstly we will define the corona product and other necessary notions. Next, we will give the explicit major formula which counts the number of spanning trees in corona product graph of two planar graphs. Then, we will propose the corona product of a planar graph and certain families of outerplanar graphs such as the Fan graph, the Star and the complete binary tree, for which we will calculate the number of spanning trees. Mathematics Subject Classification: 05C85, 05C30

Every graph G with vertex V and edge E usually referred to as G = (V, E) in this research is a simple, undirected, and non-trivial graph. A set of vertice D of graph G = (V, E) is called Dominating Set if every vertex from is adjacent to a vertex . In this paper, we analyze the relation of domination number of corona product graphs and edge corona product graphs . The difference between corona product and edge corona product is located on the placement. If on corona product, copy of H graph is located at each vertex of graph G, while at edge corona product, copy of H graph is located on each edge of graph G. The difference in the number of vertices and edges between corona product and edge corona product is and . The result of this research is . There is a relation between all domination numbers that have been generated.

The Y-index of a graph is defined by the sum of four of degrees of the vertices of a graph. Among the all topological indices the Zagreb indices have been used more considerably than any other topological indices in chemical literature. The concept of Corona Product is a recent inclusion to mathematical vocabulary. One of the most significant graph operations is the corona product of several generic and specific graphs, which is one of the most well-known graph products. In this study, we derive some explicit formulations of several corona product types, including subdivision-vertex corona, subdivision-edge corona, subdivision-vertex neighborhood corona, subdivision-edge neighborhood corona, and vertex-edge corona of two graphs.

Objectives: To examine rooted products graph and corona product of path graph with itself and cycle graph with itself for the existence of d-lucky labeling. Methods: In this study, d-lucky number for Rooted product graph of path graph to path graph (Pn ◦ Pn) and Corona product graph (Pn⊙ Pn) are computed. Method of construction is used throughout this paper to prove the theorems. Findings: Rooted products and corona products of path with itself and cycle graph with itself admit d-lucky labeling and d-lucky numbers for the same are obtained. Novelty: d-lucky numbers for some graphs are obtained by some authors but for rooted product of path with itself and corona products of path with itself and cycle with itself are new findings. Keywords: Proper Lucky labeling; Rooted product; corona product; d-lucky; d-lucky labeling

The game chromatic number $\chi_g$ is investigated for Cartesian
 product $G\square H$ and corona product $G\circ H$ of two graphs $G$
 and $H$. The exact values for the game chromatic number of Cartesian
 product graph of $S_{3}\square S_{n}$ is found, where $S_n$ is a
 star graph of order $n+1$. This extends previous results of
 Bartnicki et al. [1] and Sia [9] on the game chromatic
 number of Cartesian product graphs. Let $P_m$ be the path graph on
 $m$ vertices and $C_n$ be the cycle graph on $n$ vertices. We have
 determined the exact values for the game chromatic number of corona
 product graphs $P_{m}\circ K_{1}$ and $P_{m}\circ C_{n}$.

In the Maker-Breaker domination game, Dominator and Staller play on a graph G by taking turns in which each player selects a not yet played vertex of G. Dominator’s goal is to select all the vertices in a dominating set, while Staller aims to prevent this from happening. In this paper, the game is investigated on corona products of graphs. Its outcome is determined as a function of the outcome of the game on the second factor. Staller-Maker-Breaker domination numbers are determined for arbitrary corona products, while Maker-Breaker domination numbers of corona products are bounded from both sides. All the bounds presented are demonstrated to be sharp. Corona products as well as general graphs with small (Staller-)Maker-Breaker domination numbers are described.

Let be a nontrivial and connected graph of vertex set and edge set . A bijection is called a local edge antimagic labeling if for any two adjacent edges and , where for . Thus, the local edge antimagic labeling induces a proper edge coloring of G if each edge e assigned the color . The color of each an edge e = uv is assigned bywhich is defined by the sum of label both and vertices and . The local edge antimagic chromatic number, denoted by is the minimum number of colors taken over all colorings induced by local edge antimagic labeling of . In our paper, we present the local edge antimagic coloring of corona product of path and cycle, namely path corona cycle, cycle corona path, path corona path, cycle corona cycle.Keywords: Local antimagic; edge coloring; corona product; path; cycle.

iv 1 Preliminaries 1 2 Odd Open Dominating Sets in the Direct Product of Graphs 4 2.1 Odd Open Dominating Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Direct Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 3 Odd Closed r-Dominating Sets in Strong Products of Graphs 11 3.1 Odd Closed r-Dominating Sets . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 The Strong Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 4 The Problem of Enumeration 20 Bibliography 24 Vita 26 Abstract PARITY DOMINATION IN PRODUCT GRAPHS By Christopher Alan Whisenant, Master of Science. A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University. Virginia Commonwealth University, 2011. Director: Dewey T. Taylor, Associate Professor, Department of Mathematics and Applied Mathematics.PARITY DOMINATION IN PRODUCT GRAPHS By Christopher Alan Whisenant, Master of Science. A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science at Virginia Commonwealth University. Virginia Commonwealth University, 2011. Director: Dewey T. Taylor, Associate Professor, Department of Mathematics and Applied Mathematics. An odd open dominating set of a graph is a subset of the graph’s vertices with the property that the open neighborhood of each vertex in the graph contains an odd number of vertices in the subset. An odd closed r-dominating set is a subset of the graph’s vertices with the property that the closed r-ball centered at each vertex in the graph contains an odd number of vertices in the subset. We first prove that the n-fold direct product of simple graphs has an odd open dominating set if and only if each factor has an odd open dominating set. Secondly, we prove that the n-fold strong product of simple graphs has an odd closed r-dominating set if and only if each factor has an odd closed r-dominating set.

Let G be a graph on n vertices and A(G), L(G), and |L|(G) be the adjacency matrix, Laplacian matrix and signless Laplacian matrix of G, respectively. The paper is essentially a survey of known results about the spectra of the adjacency, Laplacian and signless Laplacian matrix of graphs resulting from various graph operations with special emphasis on corona and graph products. In most cases, we have described the eigenvalues of the resulting graphs along with an explicit description of the structure of the corresponding eigenvectors.