Exponent and scrambling index of double alternate circular snake graphs

Abstract

A graph is primitive if it contains a cycle of odd length. The exponent of a primitive graph G, denoted by exp(G), is the smallest positive integer k such that for each pair of vertices u and v in G there is a uv-walk length k. The scrambling index of a primitive graph G, denoted by k(G), is the smallest positive integer k such that for each pair of vertices u and v in G there is a uv-walk of length 2k. For an even positive integer n and an odd positive integer r, a (n,r)-double alternate circular snake graph, denoted by DA(Cr,n), is a graph obtained from a path u1u2 ... un by replacing each edge of the form u2iu2i+1 by two different r-cycles. We study the exponent and scrambling index of DA(Cr,n) and show that exp(DA(Cr,n)) = n + r − 4 and k(DA(Cr,n)) = (n + r − 3)/2.