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

04/04/2021

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by   Sebastian M. Cioabă, et al.

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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+1/d+3. As a corollary, we show that a spectral inequality on graph rigidity by Cioabă, Dewar, and Gu is essentially tight.