On the rank of graphs

Abstract

The properties of singular graphs obtained in a previous paper "On the construction of graphs of nullity one", lead to the characterization of graphs of small rank. The minimal conßgurations that are contained in singular graphs were identißed as "grown" from certain cores. A core of a singular graph G is a subgraph induced by the vertices corresponding to the non-zero components of an eigenvector in the nullspace of the adjacency matrix of G. In this paper it is shown that an arbitrary singular graph Z without isolated vertices has core-sizes corresponding to a minimal basis for the nullspace of A bounded below by 2 and above by r(Z) + 1, r(Z) being the rank of Z. For r(Z) greater or equal to 6, these bounds are sharp.