Attainability of the odd-connectivity spectral bounds

Determine whether equality can be attained in either of the bounds for the second-largest eigenvalue established in Theorem 1.3 for connected r-regular graphs whose vertex connectivity is odd and at most 2t−1; namely, determine whether λ₂(G)=min{ξ(r,t),ν(r,t)} or, when r≥6t², λ₂(G)=ξ(r,t) is possible.

Background

Theorem 1.3 gives a lower bound on λ₂(G) for every connected r-regular graph G with odd vertex connectivity κ(G)≤2t−1, namely λ₂(G)≥min{ξ(r,t),ν(r,t)}. Under the stronger condition r≥6t², the paper shows that this simplifies to λ₂(G)≥ξ(r,t).

The proof derives these values by minimizing eigenvalues of quotient matrices associated with balanced and one-sided clique configurations. However, the quotient-matrix optimization does not establish that an actual regular graph realizes either candidate bound. The authors explicitly leave unresolved whether equality is possible.