# On the inverse of the adjacency matrix of a graph

Alexander Farrugia; John Baptist Gauci; Irene Sciriha

Special Matrices (2013)

- Volume: 1, page 28-41
- ISSN: 2300-7451

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topAlexander 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 -

## References

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- [2] I. Gutman and O.E. Polansky, Mathematical Concepts in Organic Chemistry. Springer-Verlag, Berlin, 1986. Zbl0657.92024
- [3] W.H. Haemers, Interlacing eigenvalues and graphs. Linear Algebra Appl. 227–228 (1995), 593–616. [WoS]
- [4] I. Sciriha, M. Debono, M. Borg, P. Fowler, and B.T. Pickup, Interlacing–extremal graphs. Ars Math. Contemp. 6(2) (2013), 261–278. Zbl1290.05106

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