On the inverse of the adjacency matrix of a graph

Alexander Farrugia, John Baptist Gauci, and Irene Sciriha. "On the inverse of the adjacency matrix of a graph." Special Matrices 1 (2013): 28-41. <http://eudml.org/doc/267289>.

@article{AlexanderFarrugia2013,
abstract = {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.},
author = {Alexander Farrugia, John Baptist Gauci, Irene Sciriha},
journal = {Special Matrices},
keywords = {singular graph; adjacency matrix; nullity; SSP model; inverse of a graph; vertex–deleted subgraphs; NSSD; vertex-deleted subgraphs},
language = {eng},
pages = {28-41},
title = {On the inverse of the adjacency matrix of a graph},
url = {http://eudml.org/doc/267289},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Alexander Farrugia
AU - John Baptist Gauci
AU - Irene Sciriha
TI - On the inverse of the adjacency matrix of a graph
JO - Special Matrices
PY - 2013
VL - 1
SP - 28
EP - 41
AB - 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.
LA - eng
KW - singular graph; adjacency matrix; nullity; SSP model; inverse of a graph; vertex–deleted subgraphs; NSSD; vertex-deleted subgraphs
UR - http://eudml.org/doc/267289
ER -