ELA, Volume 16, pp. 451-462, December 2007, abstract.

A Characterization of Singular Graphs

Irene Sciriha

Characterization of singular graphs can be reduced to the 
non-trivial solutions of a system of linear homogeneous
equations Ax=0 for the 0-1 adjacency matrix A. A graph G 
is singular of nullity \eta(G) greater than or equal to 1, if 
the dimension of the nullspace ker(A) of its adjacency matrix
A is \eta(G). Necessary and sufficient conditions are determined 
for a graph to be singular in terms of admissible induced 
subgraphs.