Edge-grafting theorems on permanents of Laplacian matrices of graphs and their applications

Abstract

The trees, respectively unicyclic graphs, on n vertices with the smallest Laplacian permanent are studided. In this paper, by edge-grafting transformations, the n-vertex trees of given pipartitino having the second and the third smallest Laplacian permanent are identified. Similarily, the n-vertex bipartite unicyclic graphs of given bipartition having the first, second and third smallest Lapalacian permanent are characterized. Consequently, the n-vertex bipartite unicyclic graphs with the first, second and third smallest Laplacian permanent are determined.