通过谱半径构造最小$(k,k)$-边连通图
Minimally $(k,k)$-edge-connected graphs via spectral radius
AI总结：
本文基于Brualdi和Solheid的问题以及其固定大小的类比，研究了最小$(k,k)$-边连通图的谱半径最大图的结构特征，结合边交换方法和双特征向量技巧，解决了两个基础极值问题。
摘要：
对于$l > 1$，连通图$G$的$l$-边连通度$κ'_l(G)$定义为移除最少数量的边后图中至少有$l$个连通分支。一个图是严格$(k,l)$-边连通的，如果$κ'_l(G)\geq k$，但任何边的移除都会使$κ'_l(G-e)<k$。受Brualdi和Solheid的问题以及Brualdi和Hoffman提出的固定大小类比的启发，本文解决了这两个问题在最小$(k,k)$-边连通图中的情况。基于Hennayake等人[J. Graph Theory (2003)]的结构框架，结合边交换方法和双特征向量技巧，我们确定了所有给定阶数或大小的最小$(k,k)$-边连通图中谱半径最大的图的特征。我们的结果推广了Lou等人[Electron. J. Comb. (2023)]和Chen和Guo[Discrete Math. (2019)]对于$k=2$情况的研究。
英文摘要：
For $l > 1$, the $l$-edge-connectivity $κ'_l(G)$ of a connected graph $G$ is defined as the minimum number of edges whose removal leaves a graph with at least $l$ components. A graph is minimally $(k,l)$-edge-connected if $κ'_l(G)\geq k$ but for any edge $e\in E(G)$ satisfies that $κ'_l(G-e)< k$. Motivated by two foundational extremal problems: Brualdi and Solheid's problem [SIAM J. Algebra Discrete Methods (1986)] for graphs of fixed order: determine sharp upper bounds for the spectral radius over graph families and characterize extremal graphs; and its fixed size analogue proposed by Brualdi and Hoffman [Linear Algebra Appl. (1985)], we resolve both problems for minimally $(k,k)$-edge-connected graphs. Building on the structural framework of Hennayake, Lai, Li, and Mao [J. Graph Theory (2003)], we combine edge-switching method and double eigenvectors skill to characterize the graphs maximizing the spectral radius among all minimally $(k,k)$-edge-connected graphs of prescribed order or size. Our results generalize the $k=2$ cases established by Lou, Min, and Huang [Electron. J. Comb. (2023)] and Chen and Guo [Discrete Math. (2019)].
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
