Albanian Journal of Mathematics (ISNN: 1930-1235), Vol 3, No 1 (2009)

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Algebraic groups and small world graphs of high girth

V. Ustimenko


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 $\ge m$.

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