ALGEBRAIC GROUPS AND SMALL WORLD GRAPHS OF HIGH GIRTH

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 ${\mathrm{\Gamma }}_m\left(K\right)$ 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 $\mathrm{\Gamma }(m,F_{p^s}),s=1,2,\dots$ over $F_p$, where $p$ is prime, of girth $\ge m$.