A characterization of singular graphs

Abstract

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 η(G) ≥ 1, if the dimension of the nullspace ker(A) of its adjacency matrix A
is η(G). Necessary and sufficient conditions are determined for a graph to be singular in terms of
admissible induced subgraphs