Let $G$ be a simple undirected graph. The permanental sum of $G$ is equal to the permanent of the matrix $I+A\left(G\right)$, where $I$ is the identity matrix and $A\left(G\right)$ is an adjacency matrix of $G$. The computation of permanental sum is #P-complete. In this paper, we compute the permanental sum of graphs of small sizes and derive explicit formulae for the permanental sum of graphs of large sizes. The results from small sizes are used as evidence to support a conjecture on an upper bound of the permanental sum of a graph in terms of its size. The results from large sizes are obtained by a new technique of employing rook’s polynomial via the Principle of Inclusion and Exclusion.

让 $G$是一个简单的无向图。的永久总和$G$ 等于矩阵的永久性 $\mathrm{一世}+\mathrm{一种}（G）$， 在哪里 $\mathrm{一世}$ 是单位矩阵， $\mathrm{一种}（G）$ 是的邻接矩阵 $G$。永久和的计算是＃P-完全的。在本文中，我们计算了小尺寸图的永久性总和，并得出了大尺寸图的永久性总和的明确公式。小尺寸的结果用作证据来支持关于图的永久性总和的上限的猜想。大尺寸的结果是通过包含和排除原理通过采用鲁克多项式的新技术获得的。