On roots of Wiener polynomials of trees

2018/07/28 by Wang, Danielle
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1807.10967

Abstract

The Wiener polynomial of a connected graph G is the polynomial W(G;x) = ∑_i=1^D(G) d_i(G)xⁱ where D(G) is the diameter of G, and d_i(G) is the number of pairs of vertices 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 \mathbb C. We also prove that the maximum modulus among all Wiener roots of trees of order n ≥ 31 is between 2n-15 and 2n-16, 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.

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