The minimum restricted edge-connected graph and the minimum size of graphs with a given edge–degree

International audienceLet $G=(V(G),E(G))$ be a graph. Determining the minimum and/or maximum size $(|E(G)|)$ of graphs with some given parameters is a classic extremal problem in graph theory. For a graph $G$ and $e=uv∈E(G)$, we denote $d(e)=d(u)+d(v)−2$ the edge–degree of $e$. In this paper, we obtain a lower bound for the minimum size of graphs with a given order $n$, a given minimum degree $\delta$ and a given minimum edge–degree $2\delta+k−2$. Moreover, we characterize the extremal graphs for $k=0,1,2$. As an application, we characterize some kinds of minimum restricted edge connected graphs