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 ) .