Identities for minors of the Laplacian, resistance and distance matrices
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چکیده
It is shown that if L and D are the Laplacian matrix and the distance matrix of a tree respectively, then any minor of the Laplacian equals the sum of the cofactors of the complementary submatrix of D, upto a sign and a power of 2. An analogous, more general result is proved for the Laplacian and the resistance matrix of any graph. A similar identity is proved for graphs in which each block is a complete graph on r vertices, and for q-analogs of the matrices in case of a tree. The main tool is an identity for the minors of a matrix and its inverse.
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تاریخ انتشار 2009