Minimally $(k,k)$-edge-connected graphs via spectral radius

Abstract

For $l>1$, the $l$-edge-connectivity $\hat{I}{º}_l^{\prime }(G)$ 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 $\hat{I}{º}_l^{\prime }(G)\ge k$ but for any edge $e\in E(G)$ satisfies that $\hat{I}{º}_l^{\prime }(G-e)<k$. 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)].