On automorphisms of a distance-regular graph with intersection of arrays $\{39,30,4; 1,5,36\}$

A. K. Gutnova ; A. A. Makhnev

Vladikavkazskij matematičeskij žurnal, Tome 19 (2017) no. 2, pp. 11-17

Résumé

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 $\{39,30,4; 1,5,36\}$.