The kth Local Exponent of Doubly Symmetric Primitive Digraphs with d Loops

Abstract

Let D be a primitive digraph of order n. The exponent of a vertex x in V(D) is denoted γD(x), which is the smallest integer q such that for any vertex y, there is a walk of length q from x to y. Let V(D)={v1,v2,⋯,vn}. We order the vertices of V(D) so that γD(v1)≤γD(v2)≤⋯≤γD(vn) is satisfied. Then, for any integer k satisfying 1≤k≤n,γD(vk) is called the kth local exponent of D and is denoted by expD(k). Let DSn(d) represent the set of all doubly symmetric primitive digraphs with n vertices and d loops, where d is an integer such that 1≤d≤n. In this paper, we determine the upper bound for the kth local exponent of DSn(d), where 1≤k≤n. In addition, we find that the upper bound for the kth local exponent of DSn(d) can be reached, where 1≤k≤n.