几乎正则完全二部图的永久和

The Permanent Sum of Almost Regular Complete Bipartite Graph

摘要: 本文针对无向简单图G，探讨了其邻接矩阵A (G)与积和多项式之间的关系。具体而言，图G的积和多项式被定义为$p\left(G,x\right)=\text{per}\left(x1-A\left(G\right)\right)$ ，其中I代表单位矩阵。在此基础上，图G的永久和被定义为积和多项式π(G,x)的系数绝对值之和。本文研究重点探讨了完全二部图的永久和计算公式，并进一步刻画了部分完全正则二部图在删边操作后其子图的永久和计算方法。此外，给出了有向树与无向树的积和谱的关系。

Abstract: This paper focuses on the relationship between the adjacency matrix A(G) and the permanental polynomial for an undirected simple graph G. Specifically, the permanental polynomial of graph G is defined as $p\left(G,x\right)=\text{per}\left(x1-A\left(G\right)\right)$ , where I denotes the identity matrix. Based on this, the permanent sum of graph G is defined as the sum of the absolute values of the coefficients of the permanental polynomial π (G, x). This study primarily investigates the calculation formula for the permanent sum of complete bipartite graphs and further characterizes the calculation methods for the permanent sum of subgraphs obtained by edge deletion in some complete regular bipartite graphs. In addition, the relationship between the skew spectrum of directed trees and the spectrum of undirected trees is provided.