  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.
