Immanantal polynomials of the linear combination matrices of graphs

TL;DR

This paper investigates immanantal polynomials of linear combination matrices of graphs, providing combinatorial interpretations, bounds, solutions to open problems, and generalizations of classical theorems.

Contribution

It offers new combinatorial interpretations, bounds, and characterizations for immanantal polynomials of graph matrices, solving an open problem and extending classical theorems.

Findings

Provided a combinatorial interpretation of polynomial coefficients

Characterized the first six coefficients of the hook immanantal polynomial

Generalized Frobenius--K"onig and Laplace expansion theorems to immanants

Abstract

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--K\"onig 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.