Identities for minors of the Laplacian, resistance and distance matrices

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Identities for minors of the Laplacian, resistance and distance matrices

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Title: Identities for minors of the Laplacian, resistance and distance matrices
Author: BAPAT, RB; SIVASUBRAMANIAN, S
Abstract: It is shown that if L and D are the Laplacian 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. up to 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-analogues of such matrices of a tree. Our main tool is an identity for the minors of a matrix and its inverse. (C) 2011 Elsevier Inc. All rights reserved.
URI: http://dx.doi.org/10.1016/j.laa.2011.03.028
http://dspace.library.iitb.ac.in/jspui/handle/100/14150
Date: 2011


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