Minimal basis for a vector space with an application to singular graphs

Abstract

A graph is singular if its adjacency matrix is singular. In this note a parameter T(G), termed the core-width for a singular graph G, is defined. The weight of a vector is the number of non-zero components. To determine the core-width, the bases of the nullspace of A, the adjacency matrix of G, are ordered lexicographically according to their weight; then the core-width is obtained from a minimal basis in this ordering. The core-width is unique and a minimal basis in the nullspace of the adjacency matrix of G has a unique weight sequence. We show that each term in a minimal basis is less than or equal to the corresponding term of any other basis. Corresponding to such minimal bases, certain subgraphs of G of order T(G) are identifed.