Two problems on Laplacian ratios of trees

Abstract

The Laplacian ratio of graph $G$ is the permanent of the Laplacian matrix of $G$ divided by the product of degrees of all vertices. Brualdi and Goldwasser investigated systematically bounds of Laplacian ratios of trees. And they proposed two open problems: one is to characterize the extremal value of the Laplacian ratios of trees with given bipartition, the other is to determine the maximum value of the Laplacian ratios of trees. In this article, we give a solution of the first problem. We determine the lower bound of Laplacian ratios of trees with given bipartition, and the corresponding extremal graph is also determined. On the second problem, we give a conjecture on the upper bound of Laplacian ratios of trees. Furthermore, we also determine Laplacian ratios of some special trees that support the conjecture.