The purpose of this paper is to obtain a necessary and sufficient condition for the tensor product of two or more graphs to be connected, bipartite or eulerian. Also, we present a characterization of the duplicate graph $G 1 K_2$ to be unicyclic. Finally, the girth and the formula for computing the number of triangles in the tensor product of graphs are worked out.

For a graph G, let D(G) be the set of all strong orientations of G. The orientation number of G, denoted by d(G), is defined as min{d(D) | D ∈ D(G)}, where d(D) denotes the diameter of the digraph D. In this paper, a sufficient condition is given for d(G×Kn) = d(G×Kn), where G is a graph with d(G) two or three and × is the tensor product of graphs.

A graph G is an efficient open domination graph if there exists a subset D of V (G) for which the open neighborhoods centered in vertices of D form a partition of V (G). We completely describe efficient domination graphs among direct, lexicographic and strong products of graphs. For the Cartesian product we give a characterization when one factor is K2 and some partial results for grids, cylind...

It was recently proved that any graph satisfying ω > 2 3(∆ + 1) contains a stable set hitting every maximum clique. In this note we prove that the same is true for graphs satisfying ω ≥ 23(∆ + 1) unless the graph is the strong product of an odd hole and Kω/2. We also provide a counterexample to a recent conjecture on the existence of a stable set hitting every sufficiently large maximal clique.

The Merino-Welsh conjecture asserts that the number of spanning trees of a graph is no greater than the maximum of the numbers of totally cyclic orientations and acyclic orientations of that graph. We prove this conjecture for the class of series-parallel graphs.

A graph G is called Berge if neither G nor its complement contains a chordless cycle with an odd number of nodes. The famous Berge’s Strong Perfect Graph Conjecture asserts that every Berge graph is perfect. A chair is a graph with nodes {a, b, c, d, e} and edges {ab, bc, cd, eb}. We prove that a Berge graph with no induced chair (chair-free) is perfect or, equivalently, that the Strong Perfect...

If a strong monad M is used to deene the denotational semantics of a functional language with computations, a product operation ? : MX MY ! M(X Y) is needed to deene the semantics of pairing. Every strong monad is equipped with two standard products, which correspond to left-to-right and right-to-left evaluation. We study the algebraic properties of these standard products in general. Then we d...

Let G and H be graphs. The strong product G⊠H of graphs G and H is the graph with vertex set V (G) × V (H) and u = (u1, v1) is adjacent with v = (u2, v2) whenever (v1 = v2 and u1 is adjacent with u2) or (u1 = u2 and v1 is adjacent with v2) or (u1 is adjacent with u2 and v1 is adjacent with v2). In this paper, we study some properties of this operation. Also, we obtain lower and upper bounds for...

Calculating a product of multiple graphs has been studied in mathematics, engineering, computer science, and more recently in network science, particularly in the context of multilayer networks. One of the important questions to be addressed in this area is how to characterize spectral properties of a product graph using those of its factor graphs. While several such characterizations have alre...