Minimally $(k,k)$-edge-connected graphs via spectral radius
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2026
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| _version_ | 1866910244942118912 |
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| author | Wang, Yu Li, Dan Lin, Huiqiu |
| author_facet | Wang, Yu Li, Dan Lin, Huiqiu |
| contents | 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)]. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_21998 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Minimally $(k,k)$-edge-connected graphs via spectral radius Wang, Yu Li, Dan Lin, Huiqiu Spectral Theory 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)]. |
| title | Minimally $(k,k)$-edge-connected graphs via spectral radius |
| topic | Spectral Theory |
| url | https://arxiv.org/abs/2605.21998 |