Given a graph G, a dominating set D is a set of vertices such that any vertex not in D has at least one neighbor in D. A {k}-dominating multiset Dk is a multiset of vertices such that any vertex in G has at least k vertices from its closed neighborhood in Dk when counted with multiplicity. In this paper, we utilize the approach developed by Clark and Suen (2000) to prove a ‘‘Vizing-like’’ inequ...

Let P = G H be the cartesian product of graphs G,H. We relate the cover time COV[P ] of P to the cover times of its factors. When one of the factors is in some sense larger than the other, its cover time dominates, and can become of the same order as the cover time of the product as a whole. Our main theorem effectively gives conditions for when this holds. The probabilistic technique which we ...

Graphs which can be represented as nontrivial subgraphs of Cartesian product graphs are characterized. As a corollary it is shown that any bipartite, K2,3-free graph of radius 2 has such a representation. An infinite family of graphs which have no such representation and contain no proper representable subgraph is also constructed. Only a finite number of such graphs have been previously known.

The (k − 1)-fault diameter Dk(G) of a k-connected graph G is the maximum diameter of G− F for any F ⊂ V (G) with |F | < k. Krishnamoorthy and Krishnamurthy first proposed this concept and gave Dκ(G1)+κ(G2)(G1 ×G2) Dκ(G1)(G1)+ Dκ(G2)(G2) when κ(G1 ×G2)= κ(G1)+ κ(G2), where κ(G) is the connectivity of G. This paper gives a counterexample to this bound and establishes Dk1+k2(G1 × G2) Dk1(G1) +Dk2 ...

In this paper simple formulae are derived for calculating the number of spanning trees of different product graphs. The products considered in here consists of Cartesian, strong Cartesian, direct, Lexicographic and double graph. For this purpose, the Laplacian matrices of these product graphs are used. Form some of these products simple formulae are derived and whenever direct formulation was n...

The Cartesian product of two hamiltonian graphs is always hamiltonian. For directed graphs, the analogous statement is false. We show that the Cartesian product C,,, x C,, of directed cycles is hamiltonian if and only if the greatest common divisor (g.c.d.) d of n, and n, is a t least two and there exist positive integers d,, d, so that d, + d, = d and g.c.d. (n,, d,) = g.c.d. (n,, d,) = 1. We ...

We examine the distinguishing number of the Cartesian product of an arbitrary number of complete graphs. We show that for u1 ≤ · · · ≤ ud the distinguishing number of the Cartesian product of complete graphs of these sizes is either du d e or du 1/s d e + 1 where s = Πd−1 i=1 ui. In most cases, which of these values it is can be explicitly determined.

This article attempts to discuss the problems in building new topologies utilizing embedding. We propose to utilize the cartesian product to describe graphs as a right mathematical solution to adjacency matrix. We applied the cartesian product of two and sets of multiple dimensions in analysis. We proposed methodology of building logical topologies based on utilization of regular and symmetric ...