Algorithm to check the existence of H for a given G such that A(G)A(H) is graphical

Abstract

A matrix with entries [Formula: see text] is graphical if it is symmetric and all its diagonal entries are zero. Let [Formula: see text], [Formula: see text] and [Formula: see text] be graphs defined on the same set of vertices. The graph [Formula: see text] is said to be the matrix product of graphs [Formula: see text] and [Formula: see text], if [Formula: see text], where [Formula: see text] is the adjacency matrix of the graph [Formula: see text]. In such a case, we say that [Formula: see text] and [Formula: see text] are companions of each other. The main purpose of this paper is to design an algorithm to check whether a given graph [Formula: see text] has a companion. We derive conditions on [Formula: see text] and [Formula: see text] so that the generalized wheel graph, denoted by [Formula: see text], has a companion and also show that the [Formula: see text]th power of the path graph [Formula: see text] has no companion. Finally, we indicate a possible application of the algorithm in a problem of coloring of edges of the complete graph [Formula: see text].