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Abstract
Tutte's Matrix-Tree Theorem gives the number of spanning trees in a connected digraph by the determinant of a submatrix of the Laplacian, which is a generalization of the original Kirchhoff's Matrix-Tree Theorem associated to undirected graphs. In this paper, we obtain a new extension of Tutte's Matrix-Tree Theorem in terms of the signless Laplaicians. Moreover, we provide an elementary proof in terms of the factorization of signless Laplacian matrix into two incidence matrices. We finally generalize our result to the directed weighted graphs.
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