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 Cioabă 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 Cioabă, Dewar, and Gu is essentially tight.

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e present a known theorem with the following authors. Part 1 was proven by Hopcroft and Tarjan [10]. Part 2 is easy. Part 3 was proven by Garey, Johnson, and Stockmeyer [6]. Part 4 was proven by Appel, Haken, and Koch [2, 3].

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