Automorphisms of a Distance Regular Graph with Intersection Array {48,35,9;1,7,40}

Makhnev, A. A. , Bitkina, V. V. , Gutnova, A. K.
Vladikavkaz Mathematical Journal 2020. Vol. 22. Issue 2.

Abstract:
If a distance-regular graph \(\Gamma\) of diameter 3 contains a maximal locally regular 1-code perfect with respect to the last neighborhood, then \(\Gamma\) has an intersection array \(\{a(p+1),cp,a+1;1,c,ap\}\) or \({\{a(p+1),(a+1)p,c;1,c,ap\}}\), where \(a=a_3\), \(c=c_2\), \(p=p^3_{33}\) (Jurisic and Vidali). In the first case, \(\Gamma\) has an eigenvalue \(\theta_2=-1\) and \(\Gamma_3\) is a pseudo-geometric graph for \(GQ(p+1,a)\). If \(c=a-1=q\), \(p=q-2\), then \(\Gamma\) has an intersection array \(\{q^2-1,q(q-2),q+2;1,q,(q+1)(q-2)\}\), \(q>6\). The orders and subgraphs of fixed points of automorphisms of a hypothetical distance-regular graph with intersection array \(\{48,35,9;1,7,40\}\) (\(q=7\)) are studied in the paper. Let \(G={\rm Aut} (\Gamma)\) be an insoluble group acting transitively on the set of vertices of the graph \(\Gamma\), \(K=O_7(G)\), \(\bar T\) be the socle of the group \(\bar G=G/K\). Then \(\bar T\) contains the only component \(\bar L\), \(\bar L\) that acts exactly on \(K\), \(\bar L\cong L_2(7),A_5,A_6,PSp_4(3)\) and for the full the inverse image of \(L\) of the group \(\bar L\) we have \(L_a=K_a\times O_{7'}(L_a)\) and \(|K|=7^3\) in the case of \(\bar L\cong L_2(7)\), \(|K|=7^4\) otherwise.

Keywords: strongly regular graph, distance-regular graph, automorphism of graph.

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