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

REFERENCES

1.   Neumaier A. Strongly regular graphs with smallest eigenvalue -m. Arch. Math., 1979, vol. 33, no. 1, pp. 392–400. doi: 10.1007/BF01222774 .

2.   Kabanov V.V., Makhnev A.A., Paduchikh D.V. On strongly regular graphs with eigenvalue 2 and their extensions. Dokl. Math., 2010, vol. 81, no. 2, pp. 268–271. doi: 10.1134/S1064562410020298 .

3.   Makhnev A.A., Paduchikh D.V. Exceptional strongly regular graphs with eigenvalue 3. Dokl. Math., 2014, vol. 89, no. 1, pp. 20–23. doi: 10.1134/S1064562414010050 .

4.   Makhnev A.A. Strongly regular graphs with non-principal eigenvalue 4 and their extensions. Proc. of the F. Scorina Gomel State University, 2014, vol. 84, no. 3, pp. 84–85 (in Russian).