A proof of unimodality on the numbers of connected spanning subgraphs in an n-vertex graph with at least ... edges

Consider a connected undirected simple graph G=(V,E) with n vertices and m edges, and let N"i denote the number of connected spanning subgraphs with i(n-1@?i@?m) edges in G. Two well-known open problems are whether N"n"-"1,N"n,...,N"m is unimodal (posed by Welsh (1971) [21]), and whether it is log concave (posed by Mason (1972) [13]). Here, a sequence of real numbers a"0,a"1,...,a"m is said to be unimodal if there is an index i(0@?i@?m) such that a"0@?a"1@?...@?a"i>=a"i"+"1>=...>=a"m, and log concave if a"i^2>=a"i"-"1a"i"+"1 for all indices i(0