On pseudogeometric graphs of the~partial geometries $pG_2(4,t)$

A. A. Makhnev

Sbornik. Mathematics, Tome 187 (1996) no. 7, pp. 1045-1060

Résumé

An incidence system consisting of points and lines is called an $\alpha$-partial geometry of order $(s,t)$ if each line contains $s+1$ points, each point lies on $t+1$ lines (the lines intersect in at most one point), and for any point a not lying on a line $L$ there are exactly $\alpha$ lines passing through $\alpha$ and intersecting $L$ (this geometry is denoted by $pG_{\alpha }(s,t)$). The point graph of the partial geometry $pG_{\alpha }(s,t)$ is strongly regular with parameters: $v=(s+1)(1+st/\alpha )$, $k=s(t+1)$, $\lambda =(s-1)+(\alpha -1)t$ and $\mu =\alpha (t+1)$. A graph with the indicated parameters is called a pseudogeometric graph of the corresponding geometry. It is proved that a pseudogeometric graph of a partial geometry $pG_2(4,t)$ in which the $\mu$-subgraphs are regular graphs without triangles is the triangular graph $T(5)$, the quotient of the Johnson graph $J(8,4)$, or the McLaughlin graph.