Near automorphisms of trees with small total relative displacements

Abstract

For a permutation f of the vertex set V(G) of a connected graph G, let delta(f) (x,y)=vertical bar d(x,y) - d(f(x),f(y))vertical bar. Define the displacement delta(f) (G) of G with respect to f to be the sum of delta(f) (x,y) over all unordered pairs {x,y} of distinct vertices of G. Let pi(G) denote the smallest positive value of delta(f) (G) among the n! permutations f of V(G). In this note, we determine all trees T with pi(T) = 2 or 4.