A proof of unimodality on the numbers of connected spanning subgraphs in an nn-vertex graph with at least ⌈(3−22)n2+n−7−2222⌉ edges

Consider a connected undirected simple graph G=(V,E)G=(V,E) with nn vertices and mm edges, and let NiNi denote the number of connected spanning subgraphs with i(n−1≤i≤m) edges in GG. Two well-known open problems are whether Nn−1,Nn,…,NmNn−1,Nn,…,Nm is unimodal (posed by Welsh (1971) [21]), and whether it is log concave (posed by Mason (1972) [13]). Here, a sequence of real numbers a0,a1,…,ama0,a1,…,am is said to be unimodal   if there is an index i(0≤i≤m) such that a0≤a1≤⋯≤ai≥ai+1≥⋯≥ama0≤a1≤⋯≤ai≥ai+1≥⋯≥am, and log concave   if ai2≥ai−1ai+1 for all indices i(0