Solution to a Conjecture on the Permanental Sum †.

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 N P -complete. Furthermore, we characterize the graphs with the minimum permanental sum among all graphs of n vertices and m edges, where n + 3 ≤ m ≤ 2 n − 3 . [ABSTRACT FROM AUTHOR]