Abstract The research on the permanents of graph matrices is one of the contemporary research topic in algebraic combinatorics. Brualdi and Goldwasser characterized the upper and lower bounds of permanents of Laplacian matrices of trees. In this article, we determined the second and third minimal permanents of the Laplacian matrices of trees, and the second maximal permanent of the Laplacian matrices of trees is given. The corresponding extremal graphs are characterized. Furthermore, we determined bounds of permanents of the Laplacian matrices of non-caterpillar trees with given graph parameters. Moreover, the corresponding extremal graphs are determined.
Wu et al. (Wed,) studied this question.
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