Maximal and Extremal Singular Graphs

Irene Sciriha
University of Malta



Content: A graph $G$ is singular of nullity $\eta $ if the nullspace of its adjacency matrix ${\bf G}$ has dimension $\eta $. Such a graph contains $\eta $ cores determined by a basis, with minimum support sum, for the nullspace of ${\bf G}$. These are induced subgraphs of singular configurations, the latter occurring as induced subgraphs of nullity one of $G$. Singular configurations may be considered as the \lq atoms' of a singular graph. 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 show that the graphs of maximal nullity contain a substructure reaching maximal size.

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