On the inverse of the adjacency matrix of a graph

• Autorzy: Alexander Farrugia , John Baptist Gauci , Irene Sciriha
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

singular graph; adjacency matrix; nullity; SSP model; inverse of a graph; vertex–deleted subgraphs; NSSD

• Wydawca: De Gruyter Open
• Czasopismo: Special Matrices
• Rocznik: 2013
• Tom: 1
• Strony: 28-41

Daty

otrzymano

2013-08-25

zaakceptowano

2013-11-13

online

2013-11-29

Twórcy

autor

Alexander Farrugia

• Department of Mathematics, University of Malta

autor

John Baptist Gauci

• Department of Mathematics, University of Malta

autor

Irene Sciriha

• Department of Mathematics, University of Malta

Bibliografia

• [1] M. Fiedler, A characterization of tridiagonal matrices. Linear Algebra Appl., 2 (1969), 191–197.
• [2] I. Gutman and O.E. Polansky, Mathematical Concepts in Organic Chemistry. Springer-Verlag, Berlin, 1986.
• [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.

Identyfikatory

DOI

10.2478/spma-2013-0006