### Algebraic groups and small world graphs of high girth

*V. Ustimenko*

#### Abstract

We apply term algebraic graphs for an infinite family of graphs

for which the vertex set and the neighbourhood of each vertex are

quasiprojective varieties over the commutative ring $K$. For each

integral domain $K$ with unity of characteristic $\ne 2$ and integral $m \ge 2$ we construct an edge transitive graph $\Gamma_m (K)$ of girth $\ge m$ and diameter bounded by the constant independent on $K$. In particular, for each $m$ we have a family of algebraic small world graphs $\Gamma(m, F_{p^s})$ , $s= 1, 2, \dots$ over $F_p$, where $p$ is prime, of girth $\ge m$.

for which the vertex set and the neighbourhood of each vertex are

quasiprojective varieties over the commutative ring $K$. For each

integral domain $K$ with unity of characteristic $\ne 2$ and integral $m \ge 2$ we construct an edge transitive graph $\Gamma_m (K)$ of girth $\ge m$ and diameter bounded by the constant independent on $K$. In particular, for each $m$ we have a family of algebraic small world graphs $\Gamma(m, F_{p^s})$ , $s= 1, 2, \dots$ over $F_p$, where $p$ is prime, of girth $\ge m$.

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