Average connectivity and average edge-connectivity in graphs

Connectivity and edge-connectivity of a graph measure the difficulty of breaking the graph apart, but they are very much affected by local aspects like vertex degree. Average connectivity (and analogously, average edge-connectivity) has been introduced to give a more refined measure of the global “amount” of connectivity. In this paper, we prove a relationship between the average connectivity and the matching number in all graphs. We also give the best lower bound for the average edge-connectivity over n-vertex connected cubic graphs, and we characterize the graphs where equality holds. In addition, we show that this family has the fewest perfect matchings among cubic graphs that have perfect matchings.