The probability of connectedness of an unlabelled graph can be less for more edges

E. M. Wright
1972 Proceedings of the American Mathematical Society  
We write ß=ß(n, q) for the probability that a graph on n unlabelled nodes with q edges is connected; that is ß is the ratio of the number of connected graphs to the total number of graphs. We write N-n(n -\)¡2. For fixed n we might expect that ß would increase with q, at least nonstrictly. On the contrary, we show that, for any given integer s, we have ß(n,q + \)<ß (n,q) for N-n-s¿q^N-n and n>n0(s). We can show that ß(n,q + l)<ß(n,q) for a much longer range, but this requires much more elaborate arguments.
doi:10.1090/s0002-9939-1972-0295954-3 fatcat:mfpcjthxmraglev753hhs3xjsi