We supply a combinatorial description of any minor of the adjacency matrix of a graph. This description is then used to give a formula for the determinant and inverse of the adjacency matrix, A(G), of a graph G, whenever A(G) is invertible, where G is formed by replacing the edges of a tree by path bundles.
@article{bwmeta1.element.doi-10_2478_spma-2014-0009,
author = {R. B. Bapat and A. K. Lal and S. Pati},
title = {A formula for all minors of the adjacency matrix and an application},
journal = {Special Matrices},
volume = {2},
year = {2014},
zbl = {1291.05117},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_spma-2014-0009}
}
R. B. Bapat; A. K. Lal; S. Pati. A formula for all minors of the adjacency matrix and an application. Special Matrices, Tome 2 (2014) . http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_spma-2014-0009/
[1] R. B. Bapat. Graphs and Matrices. Hindustan Book Agency, New Delhi, (Copublished by Springer, London) 2011. | Zbl 1248.05002
[2] J. A. Bondy and U. S. R. Murty. Graph Theory with Applications. The Macmillan Press Ltd., London, 1976. | Zbl 1226.05083
[3] S. Chaiken. A Combinatorial Proof of the All Minors Matrix Tree Theorem. SIAM J alg disc meth, vol 3, no 3, 319–329, 1982. | Zbl 0495.05018
[4] D. M. Cvetkovic, M. Doob and H. Sachs. Spectra of Graphs. Academic Press, New York, 1979.
[5] F. Harary. The Determinant of the Adjacency Matrix of a Graph. SIAM Review, vol 4, 202–210, 1962. | Zbl 0113.17406
[6] In-Jae Kim and B. L. Shader. Smith normal form and acyclic matrices. J Algebraic Combin, vol 29, no 1, 63–80, 2009. | Zbl 1226.05158