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

TL;DR

This paper demonstrates the optimality of a spectral bound related to the number of edge-disjoint spanning trees in graphs by constructing specific counterexamples, and also confirms the tightness of a related spectral inequality in graph rigidity.

Contribution

It constructs regular graphs that disprove the possibility of improving the spectral bound for edge-disjoint spanning trees, confirming the bound's near-optimality.

Findings

Constructed $d$-regular graphs with fewer edge-disjoint spanning trees than the bound predicts.

Showed the spectral inequality on graph rigidity is essentially tight.

Confirmed the bound's optimality through explicit counterexamples.

Abstract

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.