Counterexamples to a Conjecture on Laplacian Ratios of Trees

Abstract

For a graph \(G\) with no isolated vertices, its Laplacian ratio is defined as \[ π(G)=\frac{\operatorname{per}(L(G))}{\prod_{v\in V(G)} d(v)}, \] 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 \(π(T)\) 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.