On Automorphism Groups of Strongly Regular Graphs

Our weekly CRI Graphs and Combinatorics Research Seminar will meet this week in a SPECIAL TIME this week: Thursday May 15, at 11:10

On Automorphism Groups of Strongly Regular Graphs

Martin Macaj (Comenius University, Bratislava, Slovakia)

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Abstract

Strongly regular graphs (briefly SRGs) might be considered as a combinatorial approximation of rank 3 graphs. While all rank 3 graphs are determined with the aid of the classification of finite simple groups, description of proper SRGs is, in general, very difficult problem. As a middle ground between the above cases we may consider the following problem: for a given parameter set of SRGs with integral spectrum describe all graphs with a prescribed (abstract) subgroup H of its automorhism group. This problem requires for solution a lot of computational efforts. The core ingredient of our strategy is to prescribe possible permutation actions of H on a putative graph Gamma, to determine all feasible collapsed adjacency matrices of Gamma with respect to such actions and further to attempt to reconstruct Gamma from collapsed matrices or to prove its non-existence. We will concentrate on the particular case where the group H under consideration is a cyclic group of prime order. We survey methods which can be used here, mention some recent results and pose a few open problem. As an illustration of the methods we show that there is no SRG(115,18,1,3) with an automorphism of order 23.

Host: Felix Goldberg