A real symmetric matrix G with zero diagonal encodes the adjacencies of the vertices of a graph G with weighted edges and no loops. A graph associated with a n × n non–singular matrix with zero entries on the diagonal such that all its (n − 1) × (n − 1) principal submatrices are singular is said to be a NSSD. We show that the class of NSSDs is closed under taking the inverse of G. We present results on the nullities of one– and two–vertex deleted subgraphs of a NSSD. It is shown that a necessary and sufficient condition for two–vertex deleted subgraphs of G and of the graph ⌈(G−1) associated with G−1 to remain NSSDs is that the submatrices belonging to them, derived from G and G−1, are inverses. Moreover, an algorithm yielding what we term plain NSSDs is presented. This algorithm can be used to determine if a graph G with a terminal vertex is not a NSSD.
@article{bwmeta1.element.doi-10_2478_spma-2013-0006,
author = {Alexander Farrugia and John Baptist Gauci and Irene Sciriha},
title = {On the inverse of the adjacency matrix of a graph},
journal = {Special Matrices},
volume = {1},
year = {2013},
pages = {28-41},
zbl = {1291.05120},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.doi-10_2478_spma-2013-0006}
}
Alexander Farrugia; John Baptist Gauci; Irene Sciriha. On the inverse of the adjacency matrix of a graph. Special Matrices, Tome 1 (2013) pp. 28-41. http://gdmltest.u-ga.fr/item/bwmeta1.element.doi-10_2478_spma-2013-0006/
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