Maximum and Minimum Spectral Radii in an Exceptional Family for Edge-Disjoint Spanning Trees
Abstract
For a connected graph G, let τ(G) denote the maximum number of pairwise edge-disjoint spanning trees, and let ρ(G) be its adjacency spectral radius. For integers n≥1, k≥2, and k≤δ≤2k−1, let 𝒢n,δ be the class of connected n-vertex graphs with minimum degree δ, and let ℒℋ2(n,k,δ)⊆𝒢n,δ be the exceptional family introduced by Chang, Li, and Zhang. For sufficiently large n, the sharp adjacency-spectral threshold for τ(G)≥ k is determined by maximizing ρ(G) over this family. Set h=δ−k. For each fixed admissible pair (k,δ) and all sufficiently large n, we determine the maximum and minimum adjacency spectral radii in ℒℋ2(n,k,δ). In the core--placement representation, let M be the missing-edge graph of the bounded core. For h≥2, the unique maximizer, up to isomorphism, satisfies M≅ K1,h∪(h+1)K1, with the exceptional edges nested on the large-clique side. Hence the matching configuration proposed in Conjecture~2 of Chang--Li--Zhang is not extremal. For every h≥1, the unique minimizer satisfies M≅ hK2∪2K1, with the 2h exceptional edges having distinct endpoints on the large-clique side. We also settle the cases h=0,1 and all equality cases. The proof combines an exact core--placement parametrization with a uniform Schur-complement resolvent expansion. The first candidate-dependent coefficient is affine in ∑x∈ V(M)dM(x)2, while the first placement-sensitive coefficient is a squared-load functional. Equitable quotient matrices yield the two extremal radii, and the maximizing graph gives the sharp global adjacency-spectral threshold.
Related
- Spectral radius and edge-disjoint spanning trees of graphs with edge connectivity
- Some results on the distance spectral radius and edge-disjoint spanning trees of graphs
- Minimum Distance Spectral Radius of Tree with Given Maximum Degree
- Hamiltonian graphs with prescribed minimum degree and no near-spanning cycles
- Spanning Trees with Many Leaves in Graphs of Minimum Degree at Least 7
BibTeX
@misc{cao2026maximum,
title = {{Maximum and Minimum Spectral Radii in an Exceptional Family for Edge-Disjoint Spanning Trees}},
author = {Xuanyu Cao and Chunxiang Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19655},
eprint = {2609.19655},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19655},
}
RIS
TY - UNPB
TI - Maximum and Minimum Spectral Radii in an Exceptional Family for Edge-Disjoint Spanning Trees
AU - Xuanyu Cao
AU - Chunxiang Wang
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19655
UR - https://arxiv.org/abs/2609.19655
AB - For a connected graph G, let τ(G) denote the maximum number of pairwise edge-disjoint spanning trees, and let ρ(G) be its adjacency spectral radius. For integers n≥1, k≥2, and k≤δ≤2k−1, let 𝒢_n,δ be the class of connected n-vertex graphs with minimum degree δ, and let ℒ_ℋ^2(n,k,δ)⊆𝒢_n,δ be the exceptional family introduced by Chang, Li, and Zhang. For sufficiently large n, the sharp adjacency-spectral threshold for τ(G)≥ k is determined by maximizing ρ(G) over this family. Set h=δ−k. For each fixed admissible pair (k,δ) and all sufficiently large n, we determine the maximum and minimum adjacency spectral radii in ℒ_ℋ^2(n,k,δ). In the core--placement representation, let M be the missing-edge graph of the bounded core. For h≥2, the unique maximizer, up to isomorphism, satisfies M≅ K_1,h∪(h+1)K_1, with the exceptional edges nested on the large-clique side. Hence the matching configuration proposed in Conjecture~2 of Chang--Li--Zhang is not extremal. For every h≥1, the unique minimizer satisfies M≅ hK_2∪2K_1, with the 2h exceptional edges having distinct endpoints on the large-clique side. We also settle the cases h=0,1 and all equality cases. The proof combines an exact core--placement parametrization with a uniform Schur-complement resolvent expansion. The first candidate-dependent coefficient is affine in ∑_x∈ V(M)d_M(x)^2, while the first placement-sensitive coefficient is a squared-load functional. Equitable quotient matrices yield the two extremal radii, and the maximizing graph gives the sharp global adjacency-spectral threshold.
ER -