Classification of regular two-graphs on 36 and 38 vertices (Q2760451)

In this paper a two-graph is consider as a switching class of graphs on \(n\) vertices. The spectrum of a two-graph is the Seidel spectrum of any one of the graphs in its switching class. A two-graph is said to be regular if its spectrum contains just two distinct eigenvalues \(\rho_1,\rho_2\), where \(\rho_1>\rho_2\). The only possibilities for \(\rho_1,\rho_2\) are that either they are both odd integers, or \(\rho_1=-\rho_2=\sqrt{n-1}\) and \(n-1\) is the sum of squares of integers. Corresponding to each vertex \(a\) in the vertex set of a regular two-graph there is a graph in the switching class that has that vertex as an isolated vertex. Deleting the isolated vertex gives rise to a strongly regular graph, which is called a descendant of the regular two-graph. If \(\rho_1,\rho_2\) are odd integers and the switching class of a regular two-graph contains regular graphs, then this class contains strongly regular graphs. In this paper it is proved that there are precisely 227 regular two-graphs on 36 vertices. So, there are 32,548 strongly regular graphs with parameters \((36,20,10,12)\) and 180 graphs with parameters \((36,14,4,6)\). Similar techniques were attempted in the case of regular two-graphs on 38 vertices. As a result, the known number of such regular two-graphs increases from 11 to 191.