图的邻接矩阵第n-1 Immanantal多项式的零度

The Nullity of the n-1 Immanantal Polynomial of the Adjacent Matrix of the Graph

摘要: 令S_n表示n阶对称群，λ是n的一个划分，xλ是S_n的不可约特征标，M =(m_ij)表示n阶方阵，矩阵M 的d_n-1immanantal多项式定义为$$d_{n-1}(M)=\sum _{\sigma \in S_n}{\chi }_{n-1}(\sigma )\prod _{i=1}^n{m}_{i,\sigma (i)}$$ 令图G是一个n个顶点的简单连通图，A(G)为图的邻接矩阵，图的邻接矩阵的d_n-1immanantal多项式定义为d_n-1(axI -A)，图G的零度定义为η_n-1(G)，即图G的第n 一1immanantal多项式0 根的个数，本文我们首先运用了Gallai-Edmonds结构定理去描述了最大Sachs子图，同时分别给出了图的邻接矩阵的d_n-1immanantal多项式的零根和匹配的关系，作为运用,我们给出了树、圈、完全图的η_n-1(G)，最后我们进一步刻画了η_n-1(G)=0的图。

Abstract: Let S_n denote the n-th symmetric group, let λ be a partition of n, and let xλ be the irreducible character of S_n corresponding to λ.M =(m_ij) be an n×n square matrix. The _n-1immanantal polynomial of matrix M is defined as $$d_{n-1}(M)=\sum _{\sigma \in S_n}{\chi }_{n-1}(\sigma )\prod _{i=1}^n{m}_{i,\sigma (i)}$$ Let G be a simple connected graph with n vertices, and A(G) be the adjacency matrix of G. The d_n-1 immanantal polynomial of the adjacency matrix of the graph is defined as d_n-1(axI -A). The nullity of graph G is denoted as η_n-1(G), which is the number of zero roots of the (n-1)-th immanantal polynomial of G. In this paper, we first use the Gallai-Edmonds structure theorem to describe the maximum Sachs subgraph, and establish the relationship between the zero roots of the d_n-1 immanantal polynomial of the adjacency matrix of the graph and matchings. As an application, we give η_n-1(G) for trees, cycles, and complete graphs. Furthermore, we characterize the graphs with η_n-1(G)=0.