关于图的距离谱半径和边不相交生成树的一些结果
Some results on the distance spectral radius and edge-disjoint spanning trees of graphs
- School of Mathematics and Statistics, Northwestern Polytechnical University（西北工业大学数学与统计学院）
- Xi’an-Budapest Joint Research Center for Combinatorics, Northwestern Polytechnical University（西北工业大学西安-布达佩斯组合学联合研究中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文填补了图距离谱半径与边不相交生成树数目关系中的阶数空白，确定了小阶数时的唯一极图，并在最小度条件下给出了锐利谱半径条件及极图刻画。
AI中文摘要：
设$\tau(G)$表示$n$阶连通图$G$中边不相交生成树的最大数目，并设$\rho_D(G)$表示其距离谱半径。对于整数$k\ge2$，Fan、He和Zhao [Discrete Appl. Math. 376 (2025) 31--40] 在$n\ge2k+6$时获得了保证$\tau(G)\ge k$的锐利距离谱半径条件。在本文中，我们填补了$2k\le n\le2k+5$的空白，从而完成了对所有$n\ge2k$的结果。Fan、He和Zhao给出的极图在$n\ge2k+2$时仍然有效，而我们确定了每个阶数$n=2k$和$n=2k+1$的唯一极图。我们进一步获得了在最小度条件$\delta(G)\ge k$下对所有$n\ge2k$的锐利距离谱半径条件，并刻画了所有极图。最后，对于具有更强最小度条件$\delta(G)\ge6k-4$且阶数$n\ge2\delta(G)+2$的图，我们获得了确保$\tau(G)\ge k$的锐利距离谱半径条件，并确定了唯一极图。
英文摘要：
Let $τ(G)$ denote the maximum number of edge-disjoint spanning trees in a connected graph $G$ of order $n$, and let $ρ_D(G)$ denote its distance spectral radius. For an integer $k\ge2$, Fan, He and Zhao [Discrete Appl. Math. 376 (2025) 31--40] obtained a sharp distance spectral radius condition for $τ(G)\ge k$ when $n\ge2k+6$. In this paper, we fill the gap $2k\le n\le2k+5$ and thus complete the result for all $n\ge2k$. The extremal graph given by Fan, He and Zhao remains valid for $n\ge2k+2$, while we determine the unique extremal graph for each of the orders $n=2k$ and $n=2k+1$. We further obtain sharp distance spectral radius conditions and characterize all extremal graphs under the minimum degree condition $δ(G)\ge k$ for all $n\ge2k$. Finally, for graphs with the stronger minimum degree condition $δ(G)\ge6k-4$ and order $n\ge2δ(G)+2$, we obtain a sharp distance spectral radius condition ensuring $τ(G)\ge k$ and determine the unique extremal graph.
