The \(m\)-Competition Indices of Symmetric Primitive Digraphs with Loop

Abstract

For a positive integer \(m\), where \(1 \leq m \leq n\), the \(m\)-competition index (generalized competition index) of a primitive digraph \(D\) of order \(n\) is the smallest positive integer \(k\) such that for every pair of vertices \(x\) and \(y\), there exist \(m\) distinct vertices \(v_1, v_2, \ldots, v_m\) such that there exist walks of length \(k\) from \(x\) to \(v_i\) and from \(y\) to \(v_i\) for \(1 \leq i \leq m\). In this paper, we study the generalized competition indices of symmetric primitive digraphs with loop. We determine the generalized competition index set and characterize completely the symmetric primitive digraphs in this class such that the generalized competition index is equal to the maximum value.