Minimum distinct eigenvalues of the Higman–Sims graph

Determine whether the Higman–Sims graph, the triangle-free strongly regular graph with parameters SRG(100,22,0,6), admits a real symmetric matrix described by the graph with exactly two distinct eigenvalues; equivalently, determine whether its minimum number of distinct eigenvalues satisfies q(G)=2 or q(G)=3.

Background

The paper studies the minimum number q(G) of distinct eigenvalues among real symmetric matrices whose off-diagonal zero/nonzero pattern is described by a graph G. For a non-degenerate strongly regular graph, q(G) is either 2 or 3.

The paper establishes q=3 for the Pentagon, Petersen, Hoffman–Singleton, Sims–Gewirtz, and Mesner–M22 graphs, and q=2 for the Clebsch graph. The Higman–Sims graph is the remaining member of the seven known non-complete-bipartite triangle-free strongly regular graphs, so its q-value remains unresolved. The authors also leave the same question open for any triangle-free strongly regular graph that may be discovered in the future.