Given graphs A, B and C for which A × C ∼= B × C, it is not generally true that A ∼= B. However, it is known that A × C ∼= B × C implies A ∼= B provided that C is non-bipartite, or that there are homomorphisms from A and B to C. This note proves an additional cancellation property. We show that if B and C are bipartite, then A×C ∼= B × C implies A ∼= B if and only if no component of B admits an...

Given a connected bipartite graph G, we describe a procedure which enumerates and computes all graphs H (if any) for which there is a direct product factorization G ∼= H × K2. We apply this technique to the problems of factoring even cycles and hypercubes over the direct product. In the case of hypercubes, our work expands some known results by Brešar, Imrich, Klavžar, Rall, and Zmazek [Finite ...

Figueroa-Centeno et al. introduced the following product of digraphs: let D be a digraph and let Γ be a family of digraphs such that V (F ) = V for every F ∈ Γ. Consider any function h : E(D) −→ Γ. Then the product D ⊗h Γ is the digraph with vertex set V (D) × V and ((a, x), (b, y)) ∈ E(D ⊗h Γ) if and only if (a, b) ∈ E(D) and (x, y) ∈ E(h(a, b)). In this paper, we deal with the undirected vers...

We prove that if the direct product of two connected bipartite graphs has isomorphic components, then one of the factors admits an automorphism that interchanges its partite sets. This proves a conjecture made by Jha, Klavžar and Zmazek in 1997 [P. Jha, S. Klavzar, B. Zmazek, Isomorphic components of Kronecker product of bipartite graphs, Discussiones Mathematicae Graph Theory 17 (1997) 302–308...

In this article we give a new definition of direct product of two arbitrary fuzzy graphs. We define the concepts of domination and total domination in this new product graph. We obtain an upper bound for the total domination number of the product fuzzy graph. Further we define the concept of total α-domination number and derive a lower bound for the total domination number of the product fuzzy ...

Let G and H be two graphs. The corona product G o H is obtained by taking one copy of G and |V(G)| copies of H; and by joining each vertex of the i-th copy of H to the i-th vertex of G, i = 1, 2, …, |V(G)|. In this paper, we compute PI and hyper–Wiener indices of the corona product of graphs.

 In this paper, we introduce a suitable generalization of Cayley graphs that is defined over polygroups (GCP-graph) and give some examples and properties. Then, we mention a generalization of NEPS that contains some known graph operations and apply to GCP-graphs. Finally, we prove that the product of GCP-graphs is again a GCP-graph.

Betweenness centrality is a distance-based invariant of graphs. In this paper, we use lexicographic product to compute betweenness centrality of some important classes of graphs. Finally, we pose some open problems related to this topic.