On the excess of vertex-transitive graphs of given degree and girth

We consider a restriction of the well-known Cage Problem to the class of vertex-transitive graphs, and consider the problem of finding the smallest vertex-transitive $k$-regular graphs of girth $g$. Counting cycles to obtain necessary arithmetic conditions on the parameters $(k,g)$, we extend previous results of Biggs, and prove that, for any given excess $e$ and any given degree $k\ge 4$, the asymptotic density of the set of girths $g$ for which there exists a vertex-transitive $(k,g)$-cage with excess not exceeding $e$ is 0.