Locally $GQ(3,5)$-graphs and geometries with short lines

A. A. Makhnev

Diskretnaya Matematika, Tome 10 (1998) no. 2, pp. 72-86.

Résumé

An incidence system consisting of points and lines is called the $\alpha$-partial geometry of order $(s,t)$ denoted by $pG_{\alpha}(s,t)$, if every line contains $s+1$ points, every point lies on $t+1$ lines (lines intersect in no more than one point), and for each point $a$ that does not belong to a line $L$ there exist exactly $\alpha$ lines passing through $a$ and intersecting $L$. The geometry $pG_1(s,t)$ is referred to as the generalized quadrangle, and is denoted by $GQ(s,t)$. We prove that a connected locally $GQ(3,5)$-graph is an antipodal graph of diameter three on 160 vertices. As a consequence, we obtain a classification of homogeneous extensions of partial geometries with short lines ($s\le 3$). This work was supported by the Russian Foundation for Basic Research, grant 96-01-00488.