On Automorphisms of a Distance-Regular Graph with Intersection of Arrays {39,30,4; 1,5,36}

Gutnova A. K. , Makhnev A. A.
Vladikavkaz Mathematical Journal 2017. Vol. 19. Issue 2.

Abstract:
J. Koolen posed the problem of studying distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with the second eigenvalue \(\leq t\) for a given positive integer \(t\). This problem is reduced to the description of distance-regular graphs in which neighborhoods of vertices are strongly regular graphs with non-principal eigenvalue \(t\) for \(t =1,2,\ldots\) Let \(\Gamma\) be a distance regular graph of diameter \(3\) with eigenvalues \(\theta_0>\theta_1>\theta_2>\theta_3\). If \(\theta_2= -1\), then by Proposition 4.2.17 from the book "Distance-Regular Graphs" (Brouwer A.E., Cohen A.M., Neumaier A.) the graph \(\Gamma_3\) is strongly regular and \(\Gamma\) is an antipodal graph if and only if \(\Gamma_3\) is a coclique. Let \(\Gamma\) be a distance-regular graph and the graphs \(\Gamma_2\), \(\Gamma_3\) are strongly regular. If \(k <44\), then \(\Gamma\) has an intersection array \(\{19,12,5;
1,4,15\}\), \(\{35,24,8; 1,6,28\}\) or \(\{39,30,4; 1,5,36\}\). In the first two cases the graph does not exist according to the works of Degraer J. "Isomorph-free exhaustive generation algorithms for association schemes" and Jurisic A., Vidali J. "Extremal 1-codes in distance-regular graphs of diameter 3". In this paper we found the possible automorphisms of a distance regular graph with an array of intersections {39,30,4; 1,5,36}.

Keywords: regular graph, symmetric graph, distance-regular graph, automorphism groups of graph.

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