The limiting energy conjecture for Barabási–Albert trees

Let $\{X_n\}_{n>0}$ be a sequence of trees with parameter $\alpha$, and let $\mathcal{E}(X_n)$ denote the graph energy of $X_n$.

Limiting energy conjecture. For every $\alpha$, there exists $g(\alpha)$ such that, almost surely,

$$\lim_{n\to\infty}\frac{\mathcal{E}(X_n)}{n}=g(\alpha).$$

Moreover, $g(\alpha)>1$ for $\alpha<0.79$, $g(\alpha)<1$ for $\alpha>0.81$, $g(\alpha)$ is strictly decreasing and continuous in $\alpha$, $g(\alpha)\to0$ as $\alpha\to\infty$, and $g(\alpha)\to1.273$ as $\alpha\to\infty$.

These claims formalize simulation observations about the asymptotic energy density of the random trees. In particular, the claim at $\alpha=0$ would imply that random recursive random trees are not hypoenergetic. The final two limiting assertions are internally inconsistent as written, since both use $\alpha\to\infty$; the source should be checked for a likely typographical error.