图的线性组合矩阵的immanantal多项式
Immanantal polynomials of the linear combination matrices of graphs
AI总结：
本文研究图的度数矩阵与邻接矩阵线性组合矩阵的immanantal多项式，提供系数的组合解释并推导出界，解决Merris提出的问题，推导了钩immanantal多项式的前六项系数及正则图的条件，推广Frobenius-König定理和Laplace展开定理。
摘要：
本文研究图的度数矩阵与邻接矩阵线性组合矩阵的immanantal多项式。首先，应用一般图的顶点定向概念，提供该类矩阵immanantal多项式系数的组合解释，并推导出这些系数的界，这些界隐含了Chan和Lam关于树和双图结果。此外，解决Merris提出的问题。其次，刻画了钩immanantal多项式的前六个系数，并证明了两个正则图的线性组合矩阵具有相同钩immanantal多项式的充要条件。第三，将Frobenius-König定理和Laplace展开定理推广到immanants。利用这两个定理，证明图的星度总是其线性组合矩阵immanantal多项式某根的乘率下界。最后，推导了若干重要图矩阵的钩immanantal多项式前六项系数的公式。
英文摘要：
The immanant is a generalized matrix function associated with irreducible characters of the symmetric group. The immanantal polynomial of matrix $M$ is defined as the immanant of matrix $(xI - M)$, where $I$ is the identity matrix. The characteristic polynomials and permanental polynomials of graph matrices (adjacency matrix, Laplacian matrix, signless Laplacian matrix and $A_α(G)$ matrix) have been extensively studied, yielding fascinating results. As a unification of the characteristic polynomial and permanental polynomial, the immanantal polynomial has also attracted attention and led to several important findings. In this paper, we focus on the study of immanantal polynomials for linear combination matrices composed of the degree matrix and adjacency matrix of a graph. First, applying the concept of vertex orientation for general graphs, we provide a combinatorial interpretation of the coefficients of the immanantal polynomials for the linear combination matrices of graphs, and we also characterize the bounds of these coefficients. These bounds implicitly encompass the existing results of Chan and Lam on trees and bipartite graphs. Furthermore, we give a solution to the open problem posed by Merris. Second, we characterize the first six coefficients of the hook immanantal polynomial. And the necessary and sufficient condition under which the linear combination matrices of two regular graphs have the same hook immanantal polynomial is proved. Third, we generalize the Frobenius--König theorem and the Laplace expansion theorem to immanants. Using these two theorems, we show that the star degree of a graph is always a lower bound for the multiplicity of a certain root of the immanantal polynomial of its linear combination matrix. Finally, we derive formulas for the first six coefficients of the hook immanantal polynomial for several important graph matrices.
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