On roots of Wiener polynomials of trees

Abstract

The Wiener polynomial of a connected graph G is the polynomial W(G;x)=∑i=1D(G)di(G)xi where D(G) is the diameter of G, and di(G) is the number of pairs of vertices of G at distance i from each other. We examine the roots of Wiener polynomials of trees. We prove that the collection of real Wiener roots of trees is dense in (−∞,0], and the collection of complex Wiener roots of trees is dense in ℂ. We also prove that the maximum modulus among all Wiener roots of trees of order n≥31 is between 2n−16 and 2n−15, and we determine the unique tree that achieves the maximum for n≥31. Finally, we find trees of arbitrarily large diameter whose Wiener roots are all real.