A.Kh. Zhurtov. Exceptional pseudogeometric graphs with eigenvalue r

A. Neumaier enumerated the parameters of strongly regular graphs with smallest eigenvalue $-m$. As a corollary it is proved that for a positive integer $r$ there exist only finitely many pseudogeometric graphs for $pG_{s-r}(s,t)$ with parameters different from the parameters of the net $pG_{s-r}(s,s-r)$ and from the parameters of the $pG_{s-r}(s,(s-r)(r+1)/r)$ graph complementary to the line graph of a Steiner 2-design ($s$ is a multiple of $r$). In this paper we explicitly specify functions $f(r)$ and $g(r)$ such that for $s>f(r)$ or $t>g(r)$ any pseudogeometric graph for $pG_{s-r}(s,t)$ has parameters of the net $pG_{s-r}(s,s-r)$ or parameters of $pG_{s-r}(s,(s-r)(r+1)/r)$.

Keywords: strongly regular graph, pseudogeometric graph.

The paper was received by the Editorial Office on July 5, 2018.

Archil Khazeshovich Zhurtov, Dr. Phys.-Math. Sci., Kabardino-Balkarian State University named after H.M.Berbekov, Nal’chik, 360004 Russia, e-mail: zhurtov_a@mail.ru


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