Summary: Characterization of singular graphs can be reduced to the non-trivial solutions of a system of linear homogeneous equations ${\bold {Ax=0}}$ for the 0-1 adjacency matrix ${\bold A}$. A graph $G$ is singular of nullity $\eta(G)$ greater than or equal to 1, if the dimension of the nullspace ${ker}({\bold A})$ 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.

05C50, 05C60, 05B20

adjacency matrix, eigenvalues, singular graphs, core, periphery, singular configuration, minimal configuration