I am looking for existing constructions of [vertex-symmetric graphs](https://en.wikipedia.org/wiki/Vertex-transitive_graph) on $n$ nodes that have a girth at least $g$ and are dense, i.e., have at least $n^{1 + \epsilon}$ edges, where $\epsilon>0$ may depend on $g$: in particular, if $g = O(1)$, then $\epsilon$ must be a constant bounded away from $0$ (w.r.t. $n$).


I am aware of [Cayley graphs](https://en.wikipedia.org/wiki/Cayley_graph), which are vertex-transitive, but I did not find any high-girth constructions that give *dense* graphs.



Dense vertex-symmetric graphs with high girth
If you want an explicit construction (that does not involve randomness), you can use classical constructions of Ramanujan graphs, such as the one from [this paper of Morgenstern](https://doi.org/10.1006/jctb.1994.1054).


Given an odd prime power $q$ and an even integer $d$, Theorem 4.13 in that paper gives you a Cayley graph on $n\sim q^{3d}/2$ vertices, with degree $q+1$, and girth $g$ at least (roughly) $2d$. So the number of edges is (roughly) $n^{1+3/2g}$.


