Relationship between spanning-tree packing number and graph eigenvalues

Determine the relationship between the spanning-tree packing number τ(G) and the eigenvalues of a nontrivial graph G.

Background

The paper introduces τ(G) as the maximum number of edge-disjoint spanning trees in a connected graph G and discusses spectral conditions that guarantee lower bounds on τ(G). Seymour proposed the broader problem of determining how τ(G) is related to the eigenvalues of G. The cited literature establishes several sufficient eigenvalue conditions for the existence of k edge-disjoint spanning trees, but the general relationship posed in Problem 1.1 is not resolved by those results or by the present paper, which focuses instead on sufficient conditions for the stronger property P(k, δ).

References

Seymour proposed the following problem (in private communication to Cioabă) relating τ (G) and eigenvalues of G. Problem 1.1. [3] Let G be a nontrivial graph. Determine the relationship between τ (G) and eigenvalues of G.

— Eigenvalue conditions implying edge-disjoint spanning trees and a forest with constraints  (2502.19461 - Cai et al., 26 Feb 2025) in Section 1, Introduction, Problem 1.1, p. 2

In addition, the minimum degree bound $\delta\ge6k-4$ in Theorem \ref{thm:strong} may not be best possible. Determining a better minimum degree bound is left for future work.

— Some results on the distance spectral radius and edge-disjoint spanning trees of graphs  (2609.25794 - Gao et al., 22 Sep 2026) in Section 6, Concluding remarks