Bapat, R. B. ; Sivasubramanian, Sivaramakrishnan (2011) Identities for minors of the Laplacian, resistance and distance matrices Linear Algebra and its Applications, 435 (6). pp. 1479-1489. ISSN 0024-3795
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Official URL: http://www.sciencedirect.com/science/article/pii/S...
Related URL: http://dx.doi.org/10.1016/j.laa.2011.03.028
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.
| Item Type: | Article |
|---|---|
| Source: | Copyright of this article belongs to Elsevier Science. |
| Keywords: | Laplacian; Distance Matrix; Resistance Matrix; Determinant; Partitioned Matrix; q-analogue |
| ID Code: | 81599 |
| Deposited On: | 07 Feb 2012 05:14 |
| Last Modified: | 07 Feb 2012 05:14 |
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