Explicit Group-Theoretical Constructions of Combinatorial Schemes and Their Application to the Design of Expanders and Concentrators

For every prime $p$, we construct an infinite sequence $\{Y_m\}$ of finite undirected regular graphs of degree $p+1$ such that
$$ c(Y_m)\ge(4/3+o(1))\log_pn(\mathbf Y_m), $$
where $c(X)$ and $n(X)$ are respectively the girth and the number of vertices in the graph $X$. We show that the graphs $\{Y_m\}$ may be applied for explicit construction of expanders and concentrators. Similar constructions are noted for regular graphs of degree $p^l+1$, where $p$ is prime and $l\ge 1$ is a natural number.