## Abstract

Let *G* be a connected graph with minimum degree *δ* and edge-connectivity *λ*. A graph is maximally edge-connected if *λ* = *δ*, and it is super-edgeconnected if every minimum edge-cut is trivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree. The clique number *ω*(*G*) of a graph *G* is the maximum cardinality of a complete subgraph of *G*. In this paper, we show that a connected graph *G* with clique number *ω*(*G*) ≤ *r* is maximally edge-connected or super-edge-connected if the number of edges is large enough. These are generalizations of corresponding results for triangle-free graphs by Volkmann and Hong in 2017.