Sharp spectral bipartiteness bound for regular graphs

Determine the sharp upper bound for the spectral bipartiteness invariant sigma(G)=(lambda_1(G)+lambda_n(G))/|V(G)| among all regular graphs that contain neither triangles nor pentagons.

Background

The paper completely classifies the first three extremal values of sigma(G) within the narrower class of nonbipartite distance-regular graphs of diameter three and odd girth at least seven. In particular, the folded 7-cube has value 1/32, and uniform independent blow-ups preserve this value while generally leaving the distance-regular class.

The authors explicitly state that the corresponding sharp bound for all regular graphs excluding triangles and pentagons is unresolved. Solving this problem would extend the distance-regular extremal classification to a substantially broader graph class and would determine whether graphs outside the distance-regular setting can exceed the folded 7-cube benchmark.