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.

翻译：对于无孤立顶点的图 \(G\)，其拉普拉斯比率定义为 \[ π(G)=\frac{\operatorname{per}(L(G))}{\prod_{v\in V(G)} d(v)}, \] 其中 \(L(G)\) 是 \(G\) 的拉普拉斯矩阵，\(d(v)\) 是顶点 \(v\) 的度数，\(\operatorname{per}\) 表示积和式。Brualdi 和 Goldwasser 提出了在具有固定顶点数的树 \(T\) 中最大化 \(π(T)\) 的问题。Wu、Dong 和 Lai 近期对此问题提出了一个猜想性的答案。我们给出了该猜想的无穷多反例族。