Extension of the Wiener index and Wiener polynomial

Abstract

The Wiener index W (G) of a connected graph G is defined as the sum of distances between all pairs of vertices. The Wiener polynomial H (G, x) has the property that its first derivative evaluated at x = 1 equals the Wiener index, i.e. H′ (G, 1) = W (G). The hyper-Wiener polynomial H H (G, x) satisfies the condition H H′ (G, 1) = W W (G), the hyper-Wiener index of G. In this paper we introduce a new generalization W (G, y) of the Wiener index and H (G, x, y) of the Wiener polynomial. One of the advantages of our definitions is that one can handle the Wiener and hyper-Wiener index (respectively polynomial) with the same formula, i.e. W (G) = W (G, 1), W W (G) = W (G, 2), H (G, x) = H (G, x, 1) and H H (G, x) = H (G, x, 2). © 2007 Elsevier Ltd. All rights reserved.