On the coefficient of λ in the characteristic polynomial of singular graphs

Abstract

A singular graph, with adjacency matrix A and one zero eigenvalue, has a corresponding eigenvector v₀ which is related to L, the coefficient of λ of the characteristic polynomial φ(G, λ) = Det(λI-A). In this paper a simple formula is derived expressing L in terms of the norm of v₀. Furthermore it is shown that the ratio of the diagonal cofactors, which are the determinants of the adjacency matrices of the vertex-deleted subgraphs of G, can be obtained from a kernel eigenvector. The non-singular vertex-deleted subgraphs of G are characterised. Results are also obtained for singular graphs with more than one zero eigenvalue.