On well (edge) dominated and equimatchable strong product graphs

LetG=G1×G2×⋯×Gmbe the strong product of simple, finite connected graphs, and letϕ:ℕ⟶0,∞be an increasing function. We consider the action of generalized maximal operatorMGϕonℓpspaces. We determine the exact value ofℓp-quasi-norm ofMGϕfor the case whenGis strong product of complete graphs, where0<p≤1. However, lower and upper bounds ofℓp-norm have been determined when1<p<∞. Finally, we computed the lower and upper bounds ofMGϕpwhenGis strong product of arbitrary graphs, where0<p≤1.

Strong products of Kneser graphs

This paper provides new observations on the Lovász -function of graphs. These include a simple closed-form expression of that function for all strongly regular graphs, together with upper and lower bounds on that function for all regular graphs. These bounds are expressed in terms of the second-largest and smallest eigenvalues of the adjacency matrix of the regular graph, together with sufficient conditions for equalities (the upper bound is due to Lovász, followed by a new sufficient condition for its tightness). These results are shown to be useful in many ways, leading to the determination of the exact value of the Shannon capacity of various graphs, eigenvalue inequalities, and bounds on the clique and chromatic numbers of graphs. Since the Lovász -function factorizes for the strong product of graphs, the results are also particularly useful for parameters of strong products or strong powers of graphs. Bounds on the smallest and second-largest eigenvalues of strong products of regular graphs are consequently derived, expressed as functions of the Lovász -function (or the smallest eigenvalue) of each factor. The resulting lower bound on the second-largest eigenvalue of a k-fold strong power of a regular graph is compared to the Alon–Boppana bound; under a certain condition, the new bound is superior in its exponential growth rate (in k). Lower bounds on the chromatic number of strong products of graphs are expressed in terms of the order and the Lovász -function of each factor. The utility of these bounds is exemplified, leading in some cases to an exact determination of the chromatic numbers of strong products or strong powers of graphs. The present research paper is aimed to have tutorial value as well.

A variation of the channel-assignment problem is naturally modeled by L(2,1)-labelings of graphs. An L(2,1)-labeling of a graph G is an assignment of labels from {0,1,...,/spl lambda/} to the vertices of G such that vertices at distance two get different labels and adjacent vertices get labels that are at least two apart and the /spl lambda/-number /spl lambda/(G) of G is the minimum value /spl lambda/ such that G admits an L(2,1)-labeling. The /spl Delta//sup 2/-conjecture asserts that for any graph G its /spl lambda/-number is at most the square of its largest degree. In this paper it is shown that the conjecture holds for graphs that are direct or strong products of nontrivial graphs. Explicit labelings of such graphs are also constructed.

Estimation of Laplacian spectra of direct and strong product graphs

In this paper, we determine the degree distance of G x Kr0,r1,...,rn-1 and G ? Kr0,r1,...,rn-1, where x and ? denote the tensor product and strong product of graphs, respectively, and Kr0,r1,...,rn-1 denotes the complete multipartite graph with partite sets V0,V1,...,Vn-1 where |Vj| = rj, 0 ? j ? n - 1 and n ? 3. Using the formulae obtained here, we have obtained the exact value of the degree distance of some classes of graphs.

A map f : V ? {0, 1, 2} is a Roman dominating function for G if for every vertex v with f(v) = 0, there exists a vertex u, adjacent to v, with f(u) = 2. The weight of a Roman dominating function is f(V ) = ?u?v f(u). The minimum weight of a Roman dominating function on G is the Roman domination number of G. In this article we study the Roman domination number of Cartesian product graphs and strong product graphs.

The concept of edge connectivity was first proposed by K. Menger, and in communication networks and logical networks, edge connectivity can be used to measure network reliability and fault tolerance. The graph product method can be used to construct complex networks, simulate biological molecule interactions etc. At present, research on the edge connectivity of product graphs mainly focuses on the connectivity of standard product graphs, such as Cartesian product graphs, strong product graphs. The unique properties exhibited by non standard product graphs (such as semi-strong product graphs.) in practical applications are worth further exploration. The concept of semi-strong product was proposed by Mordeson and Chang Shyh, that is, for two graphs and , their semi-strong product is a graph whose vertex set is , and the edge set is defined as follows: if and are two vertices in the semi-strong product , then there is an edge between them if and only if and and are adjacent in , or and are adjacent in and and are adjacent in . And applications of the semi-strong product in fuzzy graphs, symbolic graphs, and finance have shown its broad research prospects. In this article, we mainly study the edge connectivity of semi-strong product graphs, and obtain some exact values. Furthermore, we also give an necessary and sufficient condition for a semi-strong product to be maximally edge-connected.

On optimizing edge connectivity of product graphs

On the strong metric dimension of product graphs

A defensive alliance in a graph is a set $S$ of vertices with the property that every vertex in $S$ has at most one more‎ ‎neighbor outside of $S$ than it has inside of $S$‎. ‎A defensive alliance $S$ is called global if it forms a dominating set‎. ‎The global defensive alliance number of a graph $G$ is the minimum cardinality of a global defensive alliance in $G$‎. ‎In this article we study the global defensive alliances in Cartesian product graphs‎, ‎strong product graphs and direct product graphs‎. ‎Specifically we give several bounds for the global defensive alliance number of these graph products and express them in terms of the global defensive alliance numbers of the factor graphs.

Strong resolving partitions for strong product graphs and Cartesian product graphs

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.

Burning number of graph products

Given a graph $G$ with $n$ vertices and an Abelian group $A$ of order $n$, an $A$-distance antimagic labelling of $G$ is a bijection from $V(G)$ to $A$ such that the vertices of $G$ have pairwise distinct weights, where the weight of a vertex is the sum (under the operation of $A$) of the labels assigned to its neighbours. An {$A$-distance magic labelling} of $G$ is a bijection from $V(G)$ to $A$ such that the weights of all vertices of $G$ are equal to the same element of $A$. In this paper we study these new labellings under a general setting with a focus on product graphs. We prove among other things several general results on group antimagic or magic labellings for Cartesian, direct and strong products of graphs. As applications we obtain several families of graphs admitting group distance antimagic or magic labellings with respect to elementary Abelian groups, cyclic groups or direct products of such groups.