# zbMATH — the first resource for mathematics

A characterization of singular graphs. (English) Zbl 1142.05344
Summary: Characterization of singular graphs can be reduced to the non-trivial solutions of a system of linear homogeneous equations $${\mathbf {Ax=0}}$$ for the 0-1 adjacency matrix $${\mathbf A}$$. A graph $$G$$ is singular of nullity $$\eta(G)$$ greater than or equal to 1, if the dimension of the nullspace $$\text{ker}({\mathbf A})$$ 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.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
Full Text: