Let G be a graph with n vertices and m edges. A(G) and I denote, respectively, the adjacency matrix of G and an n by n identity matrix. For a graph G, the permanent of matrix (I+A(G)) is called the permanental sum of G. In this paper, we give a relation between the Hosoya index and the permanental sum of G. This implies that the computational complexity of the permanental sum is NP-complete. Furthermore, we characterize the graphs with the minimum permanental sum among all graphs of n vertices and m edges, where n+3≤m≤2n−3.