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DOI: 10.23671/VNC.2017.2.6504 On Automorphisms of a DistanceRegular Graph with Intersection of Arrays {39,30,4; 1,5,36}
Abstract:
J. Koolen posed the problem of studying distanceregular 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 distanceregular graphs in which neighborhoods of vertices are strongly regular graphs with nonprincipal 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 "DistanceRegular 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 distanceregular 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. "Isomorphfree exhaustive generation algorithms for association schemes" and Jurisic A., Vidali J. "Extremal 1codes in distanceregular 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, distanceregular graph, automorphism groups of graph.
Language: Russian
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For citation: Gutnova A. K., Makhnev A. A. On automorphisms of a distanceregular graph with intersection of arrays {39,30,4; 1,5,36}. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 19, no. 1, pp. 1117. DOI 10.23671/VNC.2017.2.6504 ← Contents of issue 
 

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