Uniqueness and asymptotics of roots for g0 and g1

Prove the following claims about the functions g0 and g1 defined via the auxiliary function Q2(i,n,α) = i^{α+1} + (n − 2i)(n − i − 1)^{α} + i(n − 1)^{α}: (1) For every α ≥ 2, establish that the equation g0(n, α) = Q2((n − 1)/2, n, α) − Q2(1, n, α) = 0 has exactly one real root on x ∈ (3, +∞). (2) For every α > 2, establish that the equation g1(n, α) = Q2((n − 2)/2, n, α) − Q2(1, n, α) = 0 has at most one real root on x ∈ (4, +∞), and prove that if α ≥ 5 there is exactly one real root on x ∈ (4, +∞). (3) Let x0(α) and x1(α) denote the largest real roots of g0(n, α) = 0 and g1(n, α) = 0, respectively. Prove that x0(α) − x1(α) ≥ 1 for all α ≥ 2 and that lim_{α→∞} (x0(α) − x1(α)) = 1/ln 2.

Background

In Section 4, the authors analyze when the maximum of Q2(i,n,α) over i ∈ [1, ⌊(n−1)/2⌋] occurs at i = 1 versus at i = ⌊(n−1)/2⌋. They introduce the functions g0(n,α) = Q2((n−1)/2, n, α) − Q2(1, n, α) and g1(n,α) = Q2((n−2)/2, n, α) − Q2(1, n, α) to compare these two cases for odd and even n, respectively.

These comparisons underpin threshold conditions in Theorem 2.7 and Theorem 2.8 for Hamiltonicity in terms of the zeroth-order general Randić index {^0R_α}. The conjecture seeks a precise characterization of the number of real roots of g0 and g1 and the asymptotic gap between their largest roots, which would sharpen the transition behavior between the two sufficient conditions.