Inverted Distance and Inverted Wiener Index

Abstract

The Wiener index is the sum of distances between all pairs of vertices of a (connected) graph. In this paper, we define two novel graph invariants: the inverted distance and the inverted Wiener index. The inverted distance between any two different vertices u and v of a simple connected graph G is defined as: i(u, v) = D - d(u, v) + 1, where D denotes the diameter of G and d(u, v) denotes the distance of the vertices u and v. The inverted Wiener index of a simple connected graph G is defined as: IW(G) = Sigma(u not equal v) i(u, v), where the sum is taken over unordered pairs of vertices of G. We characterized maximum trees with respect to the inverted Wiener index.