Singular graphs

Abstract

Let $\Gamma$ be a simple graph on a finite vertex set $V$ and let $A$ be its adjacency matrix. Then $\Gamma$ is said to be singular if and only if $0$ is an eigenvalue of $A.$ The nullity (singularity) of $\Gamma,$ denoted by ${\rm null}(\Gamma),$ is the algebraic multiplicity of the eigenvalue $0$ in the spectrum of $\Gamma.$ In 1957, Collatz and Sinogowitz \cite{von1957spektren} posed the problem of characterizing singular graphs. Singular graphs have important applications in mathematics and science. The chemical importance of singular graphs lies in the fact that if the nullity for the molecular graph is greater than zero then the corresponding chemical compound is highly reactive or unstable. By this reason, the chemists have a great interest in this problem. The general problem of characterising singular graphs is easy to state but it seems too difficult. In this work, we investigate this problem for graphs in general and graphs with a vertex transitive group $G$ of automorphisms. In some cases we determine the nullity of such graphs. We characterize singular Cayley graphs over cyclic groups. We show that vertex transitive graphs with $|V|$ is prime are non-singular.