Bounds for scrambling index of primitive graphs

Abstract

A connected graph is primitive provided there is a positive integer m such that for each pair of vertices u and v there is a walk of length m connecting u and v. The scrambling index of a primitive graph G is the smallest positive integer k such that for each pair of vertices u and v there is a vertex w such that there exist a walk of length k connecting u and w and a walk of length k connecting v and w. For a primitive graph G with smallest cycle Cs of length s, we present an upper bound on the scrambling index of G that depends on s and the maximum distance between vertices in G and the cycle Cs. We then classify the graphs that satisfy the upper bound.