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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.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) 05B20 Combinatorial aspects of matrices (incidence, Hadamard, etc.)
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