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Maximal core size in singular graphs. (English) Zbl 1190.05084
Summary: A graph $G$ is singular of nullity $\eta $ if the nullspace of its adjacency matrix $G$ has dimension $\eta $. Such a graph contains $\eta $ cores determined by a basis for the nullspace of $G$. These are induced subgraphs of singular configurations, the latter occurring as induced subgraphs of $G$. We show that there exists a set of $\eta $ distinct vertices representing the singular configurations. We also explore how the nullity controls the size of the singular substructures and characterize those graphs of maximal nullity containing a substructure reaching maximal size.
05C50Graphs and linear algebra
05C60Isomorphism problems (graph theory)
05B20Matrices (incidence, Hadamard, etc.)
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