Singular graphs

TL;DR

This paper investigates the properties of singular graphs, especially vertex transitive and Cayley graphs, providing characterizations and nullity calculations relevant to mathematics and chemistry.

Contribution

It characterizes singular Cayley graphs over cyclic groups and shows that vertex transitive graphs with a prime number of vertices are non-singular.

Findings

Cayley graphs over cyclic groups are characterized as singular or non-singular.

Vertex transitive graphs with a prime number of vertices are proven to be non-singular.

Nullity is determined for certain classes of graphs, aiding in understanding their spectral properties.

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.