A set S of vertices of a graph G is a geodetic set if every vertex of G lies in at least one interval between the vertices of S. The size of a minimum geodetic set in G is the geodetic number of G. Upper bounds for the geodetic number of Cartesian product graphs are proved and for several classes exact values are obtained. It is proved that many metrically defined sets in Cartesian products hav...

Let G(V;E) be a graph. The common neighborhood graph (congraph) of G is a graph with vertex set V , in which two vertices are adjacent if and only if they have a common neighbor in G. In this paper, we obtain characteristics of congraphs under graph operations; Graph :::::union:::::, Graph cartesian product, Graph tensor product, and Graph join, and relations between Cayley graphs and its c...

This article proves the following result: Let G and G′ be graphs of orders n and n′, respectively. Let G∗ be obtained from G by adding to each vertex a set of n′ degree 1 neighbors. If G∗ has game coloring number m and G′ has acyclic chromatic number k, then the Cartesian product G G′ has game chromatic number at most k(k+m − 1). As a consequence, the Cartesian product of two forests has game c...

Tardif, C., Prefibers and the Cartesian product of metric spaces, Discrete Mathematics 109 (1992) 283-288. The properties of certain sets called prefibers in a metric space are used to show that the algebraic properties of the Cartesian product of graphs generalize to metric spaces.

It is well-known that all finite connected graphs have a unique prime factor decomposition (PFD) with respect to the strong graph product which can be computed in polynomial time. Essential for the PFD computation is the construction of the so-called Cartesian skeleton of the graphs under investigation. In this contribution, we show that every connected thin hypergraph H has a unique prime fact...

This paper concerns finite, edge-transitive direct and strong products, as well as infinite weak Cartesian products. We prove that the direct product of two connected, non-bipartite graphs is edge-transitive if and only if both factors are edgetransitive and at least one is arc-transitive, or one factor is edge-transitive and the other is a complete graph with loops at each vertex. Also, a stro...

A generalization of both the hierarchical product and the Cartesian product of graphs is introduced and some of its properties are studied. We call it the generalized hierarchical product. In fact, the obtained graphs turn out to be subgraphs of the Cartesian product of the corresponding factors. Thus, some well-known properties of this product, such as a good connectivity, reduced mean distanc...

‎    The Narumi-Katayama index is the first topological index defined by the product of some graph theoretical quantities. Let G be a simple graph. Narumi-Katayama index of G is defined as the product of the degrees of the vertices of G. In this paper, we define the Narumi-Katayama polynomial of G. Next, we investigate some properties of this polynomial for graphs and then, we obtain ...

Unique minimum dominating sets in the Cartesian product of a graph and a Hamming graph are considered. A characterization of such sets is given, when they exist. A necessary and sufficient condition for the existence of a unique minimum dominating set is given in the special case of the Cartesian product of a tree and multiple copies of the same complete graph.

Department of Mathematics, Periyar University, Salem 636011, India E-mail: [email protected] 2000 Mathematics Subject Classification: primary 60A05, 47A35, secondary 28D05, 37Axx Abstract:. We show that as in the case of nfold Cartesian product for n ≥ 4, even in 3-fold Cartesian product, a related component need not be full component.