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

Abstract

Let G = (V, E) be a connected graph and let phi be a permutation of V. The total relative displacement of the permutation phi in G is delta(phi) (G) = Sigma ({x,y}CV) vertical bar d(x,y) - d(phi(x),phi(y))vertical bar, 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 phi which attains the minimum value of non-zero value of delta(phi)(G) is referred to as a near-automarphism of G and a permutation phi which attains the maximum value of delta(phi)(G) is referred to as a chaotic mapping of G. In this paper, we study the maximum value of delta(phi) (G) among all permutations in paths and cycles.