GENERALIZED EXPONENTS OF PRIMITIVE SYMMETRICAL DIGRAPHS

A strongly connected digraph D of order n is primitive (aperiodic) pro vided the greatest common divisor of its directed cycle lengths equals 1. For such a digraph there is a minimum integer t, called the expone nt of D, such that given any ordered pair of vertices x and y there is a directed walk from x to y of length t. The exponent of D is the lar gest of n 'generalized exponents' that may be associated with D. If D is a symmetric digraph, then D is primitive if and only if its underly ing graph is connected and is not bipartite. In this paper we determin e the largest value of these generalized exponents over the set of pri mitive symmetric digraphs whose shortest odd cycle length is a fixed n umber r. We also characterize the extremal digraphs. Our results are c ommon generalizations of a number of related results in the literature .