On the super connectivity of Kronecker products of graphs

In this paper we present the super connectivity of Kronecker product of a general graph and a complete graph.Comment: 8 page

Spectra and the exact number of Automorphisms of Paley-type graphs of order a product of n distinct primes

Paley graphs are Cayley graphs which are circulant and strongly regular. Paley-type graph of order a product of two distinct Pythagorean primes was introduced by Dr Angsuman Das. In this paper, we extend the study of Paley-type graphs to the order a product of n distinct Pythagorean primes. We have determined its adjacency spectra and the exact number of the automorphisms

Hyperbolicity of direct products of graphs

It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G(1) x G(2) is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs).This work was supported in part by four grants from Ministerio de Economía y Competititvidad (MTM2012-30719, MTM2013-46374-P, MTM2016-78227-C2-1-P and MTM2015-69323-REDT), Spain

A Heuristic for Direct Product Graph Decomposition

In this paper we describe a heuristic for decomposing a directed graph into factors according to the direct product (also known as Kronecker, cardinal or tensor product). Given a directed, unweighted graph G with adjacency matrix Adj(G), our heuristic aims at identifying two graphs G 1 and G 2 such that G = G 1 × G 2 , where G 1 × G 2 is the direct product of G 1 and G 2 . For undirected, connected graphs it has been shown that graph decomposition is “at least as difficult” as graph isomorphism; therefore, polynomial-time algorithms for decomposing a general directed graph into factors are unlikely to exist. Although graph factorization is a problem that has been extensively investigated, the heuristic proposed in this paper represents – to the best of our knowledge – the first computational approach for general directed, unweighted graphs. We have implemented our algorithm using the MATLAB environment; we report on a set of experiments that show that the proposed heuristic solves reasonably- sized instances in a few seconds on general-purpose hardware. Although the proposed heuristic is not guaranteed to find a factorization, even if one exists; however, it always succeeds on all the randomly-generated instances used in the experimental evaluation

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view