Further strongly regular triangle-free graphs with odd common-neighbor parameter

Determine whether strongly regular graphs with parameters $\lambda=0$ and odd $\mu$ exist beyond the seven currently known members: the pentagon, Petersen, Clebsch, Hoffman--Singleton, Gewirtz, Mesner--M22, and Higman--Sims graphs.

Background

The paper studies connected triangle-free regular graphs whose neighborhoods may be maximum odd independent sets or whose odd independence number equals the degree. The Petersen graph and odd complete bipartite graphs provide known examples. The authors relate these examples to strongly regular graphs with λ=0\lambda=0 and odd μ\mu, noting that only seven such graphs are known and that the existence of additional members is unresolved.

References

It is a famous open problem whether srg$(n,r, \lambda=0 , \mu)$ contains further members beyond the seven known ones: namely, the pentagon, Petersen, Clebsch, Hoffman--Singleton, Gewirtz, Mesner--M22, and Higman--Sims.

— The odd independence number of graphs, I: Foundations and classical classes  (2509.20763 - Caro et al., 25 Sep 2025) in Problem 3, Section 7, “Concluding remarks and open problems”