A study of total relative displacements of permutations in paths and cycles

Abstract

Let G = (V, E) be a connected graph and let Φ be a permutation of V. The total relative displacement of the permutation Φ in G is {equation presented} where d(x, y) means the distance between x and y in G, i.e., the length of a shortest path between x and y. A permutation Φ which attains the minimum value of non-zero value of δ_Φ(G) is referred to as a near-automorphism of G and a permutation Φ which attains the maximum value of δ_Φ(G) is referred to as a chaotic mapping of G. In this paper, we study the maximum value of δ_Φ(G) among all permutations in paths and cycles.