边不相交生成树的例外族中谱半径的最大值与最小值

中文摘要 AI 辅助

对于连通图$G$，设$\tau(G)$表示两两边不相交的生成树的最大数目，并设$\rho(G)$为其邻接谱半径。对于整数$n\ge1$、$k\ge2$以及$k\le\delta\le2k-1$，令$\mathcal{G}_{n,\delta}$为具有最小度$\delta$的连通$n$顶点图的类，并令$\mathcal{L}_{\mathcal{H}}^{2}(n,k,\delta)\subseteq\mathcal{G}_{n,\delta}$为由Chang、Li和Zhang引入的例外族。对于足够大的$n$，$\tau(G)\ge k$的尖锐邻接谱阈值由在该族上最大化$\rho(G)$确定。设$h=\delta-k$。对于每个固定的可容许对$(k,\delta)$以及所有足够大的$n$，我们确定了$\mathcal{L}_{\mathcal{H}}^{2}(n,k,\delta)$中邻接谱半径的最大值和最小值。在核心-放置表示中，令$M$为有界核心的缺失边图。对于$h\ge2$，在同构意义下，唯一的最大化图满足$M\cong K_{1,h}\cup(h+1)K_1$，且例外边嵌套在大团侧。因此，Chang-Li-Zhang猜想2中提出的匹配配置不是极值的。对于每个$h\ge1$，唯一的最小化图满足$M\cong hK_2\cup2K_1$，且$2h$条例外边在大团侧具有不同的端点。我们还解决了$h=0,1$的情况以及所有相等的情况。证明结合了精确的核心-放置参数化与统一的Schur补余预解式展开。第一个依赖于候选者的系数是$\sum_{x\in V(M)}d_M(x)^2$的仿射函数，而第一个对放置敏感的系数是平方负载泛函。等商矩阵产生两个极值半径，并且最大化图给出尖锐的全局邻接谱阈值。

英文摘要

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 $n\ge1$, $k\ge2$, and $k\leδ\le2k-1$, let $\mathcal{G}_{n,δ}$ be the class of connected $n$-vertex graphs with minimum degree $δ$, and let $\mathcal{L}_{\mathcal{H}}^{2}(n,k,δ)\subseteq\mathcal{G}_{n,δ}$ be the exceptional family introduced by Chang, Li, and Zhang. For sufficiently large $n$, the sharp adjacency-spectral threshold for $τ(G)\ge k$ 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 $\mathcal{L}_{\mathcal{H}}^{2}(n,k,δ)$. In the core--placement representation, let $M$ be the missing-edge graph of the bounded core. For $h\ge2$, the unique maximizer, up to isomorphism, satisfies $M\cong K_{1,h}\cup(h+1)K_1$, 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 $h\ge1$, the unique minimizer satisfies $M\cong hK_2\cup2K_1$, 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 $\sum_{x\in V(M)}d_M(x)^2$, 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.

发表机构

• Central China Normal University（华中师范大学）

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