On a Distance-Regular Graph with an Intersection Array {35,28,6;1,2,30}

Makhnev, A. A. , Tokbaeva, A. A.
Vladikavkaz Mathematical Journal 2019. Vol. 21. Issue 2.

Abstract:
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.

Keywords: distance-regular graph, Delsarte clique, geometric graph

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