The half-transitive graphs form a curious class of graphs with some kind of intermediate symmetry that is non-trivial to achieve. More precisely, a graph is half-transitive if its symmetry group is
- transitive on vertices,
- transitive on edges, but
- not transitive on arcs (i.e. edges cannot be flipped by a symmetry).
Several such graphs are known to exist (the Holt graph is the smallest one).
I am curious about the spectral properties of such graphs, in particular, I would like to prove bounds on the multiplicity of the second-largest eigenvalue of the adjacency matrix. Ideally, I want the following:
Question: Can we prove that for every half-transitive graph $G$ of vertex-degree $d$ the multiplicity $\mu$ of the second-largest eigenvalue cannot be in the range $3,...,d$, that is, either $\mu\le 2$ or $\mu >d$.
I have only empirical evidence for this (checking the half-transitive graphs in the Mathematica library). And I am also okay with replacing the bound $\mu\le 2$ by $\mu\le \mu_0$ for some fixed but not too large $\mu_0$.