On graphs whose Wiener complexity equals their order and on Wiener index of asymmetric graphs

摘要

If u is a vertex of a graph G, then the transmission of u is the sum of distances from u to all the other vertices of G. The Wiener complexity C-W(G) of G is the number of different complexities of its vertices. G is transmission irregular if C-W(G) = n(G). It is proved that almost no graphs are transmission irregular. Let T-n1, (n2), (n3) be the tree obtained from paths of respective lengths n(1), n(2), and n(3), by identifying an end-vertex of each of them. It is proved that T-1, n(2), n(3) is transmission irregular if and only if n(3) = n(2) + 1 and n(2) is not an element of{(k(2) - 1)/2, (k(2) - 2)/2} for some k = 3. It is also proved that if T is an asymmetric tree of order n, then the Wiener index of T is bounded by (n(3) - 13n + 48)/6 with equality if and only if T = T-1,(2),(n-4). A parallel result is deduced for asymmetric uni-cyclic graphs. (C) 2018 Elsevier Inc. All rights reserved.