Extremal spectral radius of graphs with cyclic edge-connectivity

An edge-cut $F$ of a graph $G$ is a cyclic edge-cut if $G-F$ has at least two components containing cycles. The minimum cardinality over all cyclic edge-cuts of $G$ is called the cyclic edge-connectivity of $G$, denoted by $c_{\lambda }\left(G\right)$. In this paper, we establish a tight upper bound for the cyclic edge-connectivity $c_{\lambda }$ among all graphs of order $n$ with minimum degree $\delta$. Denote by $H_{n,\delta }^{c_{\lambda }}$ the set of graphs of order $n$ with cyclic edge-connectivity $c_{\lambda }$ and minimum degree $\delta$. Moreover, we determine the maximum spectral radius among all graphs in $H_{n,\delta }^{c_{\lambda }}$ and characterize the corresponding extremal graphs. This generalizes some previous results on edge-connectivity and essential edge-connectivity.