Hook immanantal equalities for linear combination matrices of (di)graphs and their applications

TL;DR

This paper explores new equalities involving hook immanants of matrices derived from graphs and digraphs, providing recursive formulas and applications in graph matrix analysis.

Contribution

It characterizes hook immanantal equalities for linear combinations of degree and adjacency matrices of graphs and digraphs, introducing new recursive formulas.

Findings

Derived hook immanantal equalities for graph matrices

Established recursive formulas for hook immanantal polynomials

Applied results to analyze graph and digraph matrices

Abstract

Let $\chi_\lambda$ be an irreducible character of the symmetric group $S_n$. For an $n \times n$ matrix $M = (m_{ij})$, define the immanant of $M$ corresponding to $\chi_\lambda$ by \begin{eqnarray*} d_\lambda(M) = \sum_{\sigma \in S_n} \chi_\lambda(\sigma) \prod_{i=1}^n m_{i\sigma(i)}. \end{eqnarray*} For $\lambda = (k, 1^{n-k})$, the immanant $d_{(k, 1^{n-k})}(M)$ is called the hook immanant and denoted by $d_k(M)$. The hook immanant polynomial of matrix $M$ is defined as $d_{k}(xI_n - M)$, where $I_n$ is the $n \times n$ identity matrix. Let $G$ and $\overrightarrow{G}$ be a graph and a digraph, respectively. Suppose that $D(G)$ and $A(G)$ (resp. $D(\overrightarrow{G})$ and $A(\overrightarrow{G})$) are the degree matrix and adjacency matrix of $G$ (resp. $\overrightarrow{G}$), respectively. In this paper, we characterize two hook immanantal equalities for the linear combination of matrices $\beta D(G)+\gamma A(G)$ and $\beta D(\overrightarrow{G})+\gamma A(\overrightarrow{G})$, where $\beta$ and $\gamma$ are real numbers. As applications, we derive recursive formulas for the hook immanantal polynomials and hook immanants of graph matrices.