On a distance-regular graph with an intersection array $\{35,28,6;1,2,30\}$

A. A. Makhnev ; A. A. Tokbaeva

Vladikavkazskij matematičeskij žurnal, Tome 21 (2019) no. 2, pp. 27-37

Résumé

It is proved that for a distance-regular graph $\Gamma$ of diameter $3$ with eigenvalue $\theta_2=-1$ the complement graph of $\Gamma_3$ is pseudo-geometric for $pG_{c_3}(k,b_1/c_2 )$. Bang and Koolen investigated distance-regular graphs with intersection arrays ${(t+1)s,ts, (s+1-\psi); 1,2,(t+1)\psi}$. If $t=4$, $s=7$, $\psi=6$ then we have array ${35,28,6;1,2,30}$. Distance-regular graph $\Gamma$ with intersection array $\{35,28,6; 1,2,30\}$ has spectrum of $35^1$, $9^{168}$, $-1^{182}$, $-5^{273}$, $v=1+35+490+98=624$ vertices and $\overline{\Gamma}_3$ is a pseudogeometric graph for $pG_{30}(35,14)$. Due to the border of Delsarte, the order of clicks in $\Gamma$ is not more than $8$. It is also proved that either a neighborhood of any vertex in $\Gamma$ is the union of an isolated $7$-click, or the neighborhood of any vertex in $\Gamma$ does not contain a $7$-click and is a connected graph. The structure of the group $G$ of automorphisms of a graph $\Gamma$ with an intersection array $\{35,28,6; 1,2,30\}$ has been studied. In particular, $\pi(G)\subseteq\{2,3,5,7,13\}$ and the edge symmetric graph $\Gamma$ has a solvable group automorphisms.