Counterexamples to a Conjecture on Laplacian Ratios of Trees

TL;DR

This paper presents infinite counterexamples to a conjecture about the maximum Laplacian ratio of trees, challenging a recent proposed answer to an open problem.

Contribution

The authors provide infinite families of counterexamples to Wu, Dong, and Lai's conjecture on Laplacian ratios of trees.

Findings

Counterexamples disprove the conjecture.

Infinite families of trees with Laplacian ratios exceeding the conjectured maximum.

Abstract

For a graph \(G\) with no isolated vertices, its Laplacian ratio is defined as \[ \pi(G)=\frac{\operatorname{per}(L(G))}{\prod_{v\in V(G)} d(v)}, \] where \(L(G)\) is the Laplacian matrix of \(G\), \(d(v)\) is the degree of \(v\), and \(\operatorname{per}\) denotes the permanent. Brualdi and Goldwasser asked for the maximum value of \(\pi(T)\) among trees \(T\) with a fixed number of vertices. Wu, Dong and Lai recently proposed a conjectural answer to this problem. We give infinite families of counterexamples to their conjecture.