Proof of a conjecture of Graham and Lov\'asz concerning unimodality of coefficients of the distance characteristic polynomial of a tree

TL;DR

This paper proves Graham and Lovász's conjecture that the normalized coefficients of a tree's distance characteristic polynomial are unimodal and log-concave, confirming their conjectured properties.

Contribution

It provides a proof that the coefficients of the distance characteristic polynomial of a tree are both unimodal and log-concave, resolving a longstanding conjecture.

Findings

Coefficients are unimodal.

Coefficients are log-concave.

Conjecture of Graham and Lovász confirmed.

Abstract

We establish a conjecture of Graham and Lov\'asz that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal; we also prove they are log-concave.