Maximal core size in singular graphs

A graph G is singular of nullity η if the nullspace of its adjacency matrix G has dimension η . Such a graph contains η 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 η 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.