A Lower Bound of the \( l \)-Edge-Connectivity and Optimal Graphs

For an integer \( l > 1 \), the \( l \)-edge-connectivity of a graph \( G \) with \( |V(G)| \geq l \), denoted by \( \lambda_l(G) \), is the smallest number of edges whose removal results in a graph with \( l \) components. In this paper, we study lower bounds of \( \lambda_l(G) \) and optimal graphs that reach the lower bounds. Former results by Boesch and Chen are extended.

We also present in this paper an optimal model of interconnection network \( G \) with a given \( \lambda_l(G) \) such that \( \lambda_2(G) \) is maximized while \( |E(G)| \) is minimized.