Let G be a simple undirected graph, I the identity matrix, and A(G) an adjacency matrix of G. Then the permanental sum of G equals to the permanent of the matrix \(I+A(G)\). Since the computation of the permanental sum of a graph is #P-complete, it is desirable to have good bounds. In this paper, we affirm a sharp upper bound for general graphs conjectured by Wu and So. Moreover, we prove a sharp lower bound for connected tricyclic graphs. Lastly, several unsolved problems about permanental sum are presented.

中文翻译：

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图的永久和的锐界

让G ^是一个简单的无向图，我单位矩阵，和甲（ģ）的邻接矩阵G ^。那么G的永久和等于矩阵\(I+A(G)\)的永久和。由于图的永久和的计算是#P-complete，因此需要有好的边界。在本文中，我们肯定了 Wu 和 So 推测的一般图的一个尖锐的上界。此外，我们证明了连接三环图的尖锐下界。最后，提出了几个关于永久金额的未解决问题。