Extremal Properties of Complex Networks
  We describe the structure of connected graphs with the minimum and maximum average distance, radius, diameter, betweenness centrality, efficiency and resistance distance, given their order and size. We find tight bounds on these graph qualities for any arbitrary number of nodes and edges and analytically derive the form and properties of such networks. 
Dionysios Barmpoutis and Richard M. Murray
q-bio.MN cond-mat.stat-mech cs.SI physics.soc-ph