The exponent of a digraph and the diameter of its multiple direct product

Abstract

A new phenomenon pertaining to the diameter of the multiple direct product D-m of a primitive digraph D is found related to exp(D). It is shown that there is a positive integer m, referred to as the critical multiplicity of D, which satisfies the condition diam(D) < diam(D-2) < center dot center dot center dot < diam(Dm-1) < diam(D-m) = diam(Dm+1) = center dot center dot center dot = exp(D). Further, it is proved that the critical multiplicity m of D satisfies m < n - 1 where n is the order of D. The extremal cases are classified as follows: for each n, there are two primitive digraphs up to isomorphism having a critical multiplicity of n - 1, where one of the digraphs is the Wielandt digraph. (C) 2012 Elsevier Inc. All rights reserved.