Exponent-critical primitive graphs and the kronecker product

Abstract

A directed graph is primitive of exponent t if it contains walks of length t between all pairs of vertices, and t is minimal with this property. Moreover, it is exponent-critical if the deletion of any arc results in an imprimitive graph or in a primitive graph with strictly greater exponent. We establish necessary and sufficient conditions for the Kronecker product of a pair of graphs to be exponent-critical of prescribed exponent, defining some refinements of the concept of exponentcriticality in the process.