Extremal problems on saturation for the family of $k$-edge-connected graphs

Let $\mathcal{F}$ be a family of graphs. A graph $G$ is $\mathcal{F}$-saturated if $G$ contains no member of $\mathcal{F}$ as a subgraph but $G+e$ contains some member of $\mathcal{F}$ whenever $e\in E(\overline{G})$\mathrm{.}Thesaturationnumberandextremalnumberof$$\mathcal{F}$$, denoted $sat(n,\mathcal{F})$ and $ex(n,\mathcal{F})$ respectively, are the minimum and maximum numbers of edges among $n$-vertex $\mathcal{F}$-saturated graphs. For $k\in \mathbb{N}$, let ${\mathcal{F}}_k$ and ${\mathcal{F}}_k^{\mathrm{\prime }}$ be the families of $k$-connected and $k$-edge-connected graphs, respectively. Wenger proved $sat(n,{\mathcal{F}}_k)=(k-1)n-\left(\genfrac{}{}{0px}{}{k}{2}\right)$, we prove $sat(n,$\mathcal{F}$‘_k)=(k-1)(n-1)-\lfloor{\frac {n}{k+1}}\rfloor{k-1 \choose 2}$\mathrm{.}Wealsoprove$ex(n,$\mathcal{F}$’_k)=(k-1)n-{k\choose2}$ and characterize when equality holds. Finally, we give a lower bound on the spectral radius for ${\mathcal{F}}_k$-saturated and ${\mathcal{F}}_k^{\mathrm{\prime }}$-saturated graphs.