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

TL;DR

This paper characterizes the maximum spectral radius of minimally $(k,k)$-edge-connected graphs, solving extremal problems related to spectral bounds and graph structure for fixed order or size.

Contribution

It provides a complete characterization of extremal minimally $(k,k)$-edge-connected graphs maximizing spectral radius, extending previous specific case results.

Findings

Identifies graphs with maximum spectral radius among minimally $(k,k)$-edge-connected graphs.

Generalizes previous results for the case $k=2$ to all $k>1$.

Combines edge-switching and eigenvector techniques for characterization.

Abstract

For $l > 1$, the $l$-edge-connectivity $\kappa'_l(G)$ 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 $\kappa'_l(G)\geq k$ but for any edge $e\in E(G)$ satisfies that $\kappa'_l(G-e)< k$. 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)].