New strongly regular graphs on 53 vertices

Source abstract

The only known strongly regular graph with parameters (53, 26, 12, 13) is the one coming from the famous Paley construction. In this paper, we provide new examples of strongly regular graphs with 53 vertices. We established that if an automorphism group of prime order p acts on such a strongly regular graph, then p ∈ {2, 3, 13, 53}. We have constructed all strongly regular graphs on 53 vertices with Z13 or Z6 as an automorphism group. We also give examples of new regular two-graphs with 54 vertices that have at least one descendant whose automorphism group is either an abelian group of order six or a cyclic group of order 13. Finally, we have proved the existence of 643 mutually nonisomorphic strongly regular graphs with parameters (53, 26, 12, 13) as their descendants.