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DOI: 10.23671/VNC.2017.2.6504

# 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.