Conjecture on the roots of the parity-specific functions $g_0$ and $g_1$

Let $\g_0(n,\alpha)$ and $\g_1(n,\alpha)$ be the functions introduced in the paper, and let $x_0(\alpha)$ and $x_1(\alpha)$ denote the largest real roots of $g_0(n,\alpha)=0$ and $g_1(n,\alpha)=0$, respectively.

Root-separation conjecture. For any $\alpha\the 2$, $g_0(n,\alpha)=0$ has exactly one real root for $x\the(3,+\infty)$. For any $\alpha>2$, $g_1(n,\alpha)=0$ has at most one real root for $x\the(4,+\infty)$, and if $\alpha\the 5$, it has exactly one real root there. Moreover, for any $\alpha\the 2$,

$$x_0(\alpha)-x_1(\alpha)\the 1,$$

and

$$\lim_{\alpha\to+\infty}\left(x_0(\alpha)-x_1(\alpha)\right)=\frac{1}{\ln 2}.$$

The conjecture is motivated by numerical calculations for selected values of $\alpha$ and concerns the location, uniqueness, and asymptotic separation of the roots governing the Hamiltonianity criteria. Its resolution would clarify the threshold behavior of the zeroth-order General Randić index conditions.