Extremality of the constructed graphs for even vertex-connectivity bounds

Prove that, for every fixed integer t≥2, there exists an integer R(t)>2t² such that, for every integer r≥R(t) satisfying t∣(r+1), the minimum possible second-largest adjacency eigenvalue among connected r-regular graphs with vertex connectivity at most 2t equals ρ*(r,t), and hence that a graph G*_{r,t} associated with a minimizing pair (k₁*,k₂*) is extremal in this class.

Background

For fixed t≥2 and r>2t² with t dividing r+1, the paper defines λ₂{min}(r,t) as the infimum of λ₂(G) over all connected r-regular graphs G satisfying κ(G)≤2t. Theorem 1.1 establishes the lower bound λ₂{min}(r,t)≥ρ(r,t).

The authors construct graphs G*_{r,t} with vertex connectivity 2t and second-largest eigenvalue ρ(r,t), yielding λ₂{min}(r,t)≤ρ(r,t) by Theorem 1.2. Remark 2.14(a) shows that ρ*(r,t) is asymptotically close to ρ(r,t) for each fixed t, but the paper does not prove that the constructed graphs attain the true infimum. The conjecture asserts their eventual extremality when r is sufficiently large and t divides r+1.

References

For every fixed integer $t\geq2$, there exists an integer $R(t)>2t2$ such that, for every integer $r\geq R(t)$ satisfying $t\mid(r+1)$, $\lambda_2{\min}(r,t)=\rho*(r,t)$. Equivalently, a graph $G*_{r,t}$ associated with a pair $(k_1,k_2^)$ achieving the minimum in~eq:rhostar is extremal among all connected $r$-regular graphs with $\kappa(G)\leq2t$.

eq:rhostar:

ρ∗(r,t)=min⁡{λmax⁡(M(t−2k1,t+1−2k2)):k1,k2∈Z,0≤k1≤t,1≤k2≤t},\rho^*(r,t)=\min\Bigl\{\lambda_{\max}\bigl(M(t-2k_1,t+1-2k_2)\bigr): k_1,k_2\in\mathbb{Z}, 0\leq k_1\leq t, 1\leq k_2\leq t\Bigr\},

— Extremal spectral gap of regular graphs with bounded vertex connectivity  (2610.02992 - Wang et al., 2 Oct 2026) in Conjecture 1.1, Section 1 (Introduction)