Regular two-graphs are antipodal distance-regular double coverings of the complete graph, and they have many interesting combinatorial properties. We derive a construction for regular two-graphs containing cliques of specified order from their connection to large sets of equiangular lines in Euclidean space. It is shown that the existence of a regular two-graph with least eigenvalue τ containing a clique of order d depends on the existence of an incidence structure on d points with special properties. Quasi-symmetric designs provide examples of these incidence structures.University of WaterlooMaster of Mathematics in Combinatorics and Optimizatio
