On the graph inequality image Original Research Article

Given a simple graph G, the mth power of G, denoted by image, has the same vertex set as G and an edge joining vertices x and y if and only if there is a walk of length less than or equal to m from x to y. Let image denote the edge clique cover number of G. In this paper, we present chordal graphs G satisfying image, which answers a question proposed by Kang et al. [Graph satisfying inequality image, Discrete Math. 250 (2002) 259–264]. We also show that image for any simple graph G and for each positive integer n, and construct a graph not satisfying image for each positive integer n. Then we give a formula for image for a tree T in terms of the number of paths of length image of certain type and the number of pendent vertices of T. Finally we present some open problems.