A formula for all minors of the adjacency matrix and an application

R. B. Bapat; A. K. Lal; S. Pati

Special Matrices (2014)

  • Volume: 2, Issue: 1, page 89-98, electronic only
  • ISSN: 2300-7451

Abstract

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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.

How to cite

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R. B. Bapat, A. K. Lal, and S. Pati. "A formula for all minors of the adjacency matrix and an application." Special Matrices 2.1 (2014): 89-98, electronic only. <http://eudml.org/doc/267427>.

@article{R2014,
abstract = {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.},
author = {R. B. Bapat, A. K. Lal, S. Pati},
journal = {Special Matrices},
keywords = {Adjacency Matrix; Linear Subgraphs; Minors; Path Bundle; adjacency matrix; linear subgraphs; minors; path bundle},
language = {eng},
number = {1},
pages = {89-98, electronic only},
title = {A formula for all minors of the adjacency matrix and an application},
url = {http://eudml.org/doc/267427},
volume = {2},
year = {2014},
}

TY - JOUR
AU - R. B. Bapat
AU - A. K. Lal
AU - S. Pati
TI - A formula for all minors of the adjacency matrix and an application
JO - Special Matrices
PY - 2014
VL - 2
IS - 1
SP - 89
EP - 98, electronic only
AB - 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.
LA - eng
KW - Adjacency Matrix; Linear Subgraphs; Minors; Path Bundle; adjacency matrix; linear subgraphs; minors; path bundle
UR - http://eudml.org/doc/267427
ER -

References

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  1. [1] R. B. Bapat. Graphs and Matrices. Hindustan Book Agency, New Delhi, (Copublished by Springer, London) 2011. Zbl1248.05002
  2. [2] J. A. Bondy and U. S. R. Murty. Graph Theory with Applications. The Macmillan Press Ltd., London, 1976. Zbl1226.05083
  3. [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. Zbl0495.05018
  4. [4] D. M. Cvetkovic, M. Doob and H. Sachs. Spectra of Graphs. Academic Press, New York, 1979. 
  5. [5] F. Harary. The Determinant of the Adjacency Matrix of a Graph. SIAM Review, vol 4, 202–210, 1962. Zbl0113.17406
  6. [6] In-Jae Kim and B. L. Shader. Smith normal form and acyclic matrices. J Algebraic Combin, vol 29, no 1, 63–80, 2009. Zbl1226.05158

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