Extremal Graphs for a Spectral Inequality on Edge-Disjoint Spanning Trees

Liu, Hong, Gu, and Lai proved if the second largest eigenvalue of the adjacency matrix of graph $G$ with minimum degree $\delta \ge 2m+2 \ge 4$ satisfies $\lambda_2(G) < \delta - \frac{2m+1}{\delta+1}$, then $G$ contains at least $m+1$ edge-disjoint spanning trees, which verified a generalization of a conjecture by Cioab\u{a} and Wong. We show this bound is essentially the best possible by constructing $d$-regular graphs $\mathcal{G}_{m,d}$ for all $d \ge 2m+2 \ge 4$ with at most $m$ edge-disjoint spanning trees and $\lambda_2(\mathcal{G}_{m,d}) < d-\frac{2m+1}{d+3}$. As a corollary, we show that a spectral inequality on graph rigidity by Cioab\u{a}, Dewar, and Gu is essentially tight.