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Node degree distribution in spanning trees

A method is presented for computing the number of spanning trees involving one link or a specified group of links, and excluding another link or a specified group of links, in a network described by a simple graph in terms of derivatives of the spanning-tree generating function defined with respect to the eigenvalues of the Kirchhoff (weighted Laplacian) matrix. The method is applied to deduce the node degree distribution in a complete or randomized set of spanning trees of an arbitrary network. An important feature of the proposed method is that the explicit construction of spanning trees is not required. It is shown that the node degree distribution in the spanning trees of the complete network is described by the binomial distribution. Numerical results are presented for the node degree distribution in square, triangular, and honeycomb lattices.