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 Dm 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 conditiondiam(D)<diam(D2)<⋯<diam(Dm-1)<diam(Dm)=diam(Dm+1)=⋯=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.