An Estimation of the Shortest and Largest Average Path Length in Graphs   of Given Density
  Many real world networks (graphs) are observed to be 'small worlds', i.e., the average path length among nodes is small. On the other hand, it is somewhat unclear what other average path length values networks can produce. In particular, it is not known what the maximum and the minimum average path length values are. In this paper we provide a lower estimation for the shortest average path length (l) values in connected networks, and the largest possible average path length values in networks with given size and density. To the latter end, we construct a special family of graphs and calculate their average path lengths. We also demonstrate the correctness of our estimation by simulations. 
L\'aszl\'o Guly\'as, G\'abor Horv\'ath, Tam\'as Cs\'eri and George   Kampis
physics.soc-ph cs.SI