Laplacian Permanents of Trees

Abstract

Let T be a tree on n vertices. The Laplacian matrix $L( T )$ is the difference of the diagonal matrix of vertex degrees and the adjacency matrix. The main result of this article is that, for “almost all” trees T, there is a nonisomorphic tree $T^\prime $ such that per $L ( T ) = {\text{per }}L ( T^\prime )$. The proof follows the approach taken by Schwenk in [New Directions in the Theory of Graphs, F. Harary, ed., Academic Press, New York, 1973, pp. 275–307]. The difficulty is finding a single pair of “super” trees from which to start. The search for this pair was greatly facilitated by a new algorithm for computing Laplacian permanents of trees. This algorithm is also reported. Finally, the algorithm is used to establish inequalities for per $L ( T )$.