The Wiener polynomial of a graph

Abstract

rnThe Wiener index is a graphical invariant that has found extensive application in chemistry. We define a generating function, which we call the Wiener polynomial, whose derivative is a q-analog of the Wiener index. We study some of the elementary properties of this polynomial and compute it for some common graphs. We then find a formula for the Wiener polynomial of a dendrimer, a certain highly regular tree of interest to chemists, and show that it is unimodal. Finally, we point out a connection with the Poincar6 polynomial of a finite Coxeter group. 0 1996 John Wiley & Sons, Inc.the boiling point of paraffin. Since then, the index has been shown to correlate with a host of other properties of molecules (viewed as graphs). For more information about the Wiener index in chemistry and mathematics, see [2, 31, respectively. We wish to define and study a related generating function. If q is a parameter, then the Wiener polynomial of G is

Vertex-weighted Wiener polynomials of subdivision-related graphs

“…The (unweighted) Wiener polynomial of G is defined as P 0 (G; x) = {u,v}⊆V (G) x d (u,v) , with x a dummy variable. This coincides with definitions of Hosoya [17] and Sagan et al [23]. Some authors prefer the name Hosoya polynomial.…”

GA2 index of some graph operations

“…The Wiener index of the Cartesian product of graphs was studied in [7,20]. In [17], Klavžar, Rajapakse and Gutman computed the Szeged index of the Cartesian product graphs.…”

“…As usual, the distance between the vertices u and v of G is denoted by d G (u, v) (d(u, v) for short). It is defined as the length of a minimum path connecting them and d G (u)(d(u) for short) denotes the degree of u in G. The Wiener index of a graph G is defined as W (G) = {u,v} d (u, v) [7,17,20,23]. GA 2 index of the graph of G is defined by GA 2 (G) =…”

Wiener-type invariants of some graph operations

“…The Wiener index of the Cartesian product graphs was studied in [5,19]. In [14], Klavžar, Rajapakse and Gutman computed the Szeged index of the Cartesian product graphs.…”