A Max-Min problem on spectral radius and connectedness of graphs

In the past decades, many scholars concerned which edge-extremal problems have spectral analogues? Recently, Wang, Kang and Xue showed an interesting result on $F$-free graphs [J. Combin. Theory Ser. B 159 (2023) 20--41]. In this paper, we study the above problem on critical graphs.Let $P$ be a property defined on a family $\mathbb{G}$ of graphs. A graph $G$ in $\mathbb{G}$ is said to be $P$-critical,if it has the property $P$ but $G-e$ no longer has for any edge $e\in E(G)$. Especially, a graph is minimally $k$-(edge)-connected,if it is $k$-connected (respectively, $k$-edge connected) and deleting an arbitrary edge always leaves a graph which is not $k$-connected (respectively, $k$-edge-connected). An interesting Max-Min problem asks what is the maximal spectral radius of an $n$-vertex minimally $k$-(edge)-connected graphs? In 2019, Chen and Guo [Discrete Math. 342 (2019) 2092--2099] gave the answer for $k=2$. In 2021, Fan, Goryainov and Lin [Discrete Appl. Math. 305 (2021) 154--163] determined the extremal spectral radius for minimally $3$-connected graphs. We obtain some structural properties of minimally $k$-(edge)-connected graphs. Furthermore, we solve the above Max-Min problem for $k\geq3$, which implies that every minimally $k$-(edge)-connected graph with maximal spectral radius also has maximal number of edges. Finally, a general problem is posed for further research.