M.S. Nirova. On distance-regular graphs with  $\theta_2=-1$ ... P. 215-228

Let a distance-regular graph $\Gamma$ of diameter 3 have eigenvalue $\theta_2=-1$. Then $\Delta=\bar\Gamma_3$ is a pseudo-geometric graph for $pG_{c_3}(k,b_1/c_2)$ containing $v$ Delsarte cliques $u^\bot$ of order $k+1$. In the case $a_1=0$ we have a partition of the subgraph $\Delta(u)$ by cliques $w^\bot-\{u\}$, where $w\in \Gamma(u)$. If there exists a strongly regular graph with parameters (176,49,12,14) in which neighborhoods of vertices are $7\times 7$-lattices, then there exists a distance-regular graph with intersection array $\{7,6,6;1,1,2\}$. If $\Delta$ contains an $n$-coclique $\{u,u_2,...,u_n\}$, then there are $k_3-(n-1)(a_3+1)$ vertices in $\Gamma_3(u)-\cup_{i=2}^n \Gamma(u_i)$, which yields a new upper bound for the order of a clique in $\Gamma_3$. Moreover, it is proved that distance-regular graphs with intersection arrays $\{44,35,3;1,5,42\}$ and $\{27,20,7;1,4,21\}$ do not exist.

Keywords: distance-regular graph, eigenvalue, strongly regular graph.

The paper was received by the Editorial Office on Dezember 25, 2017.

Funding Agency:

Russian Science Foundation (Grant Number 18-11-00067).

Marina Sefovna Nirova, Cand. Sci. (Phys.-Math.), Kabardino-Balkarian State University named after H.M.Berbekov, Nal’chik, 360004 Russia, e-mail: nirova_m@mail.ru.

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