Some immanantal inequalities and equalities for linear combination matrices of (di)graphs

Let χλ be an irreducible character of the symmetric group Sn. For an n×n matrix M=(mij), define the immanant of M corresponding to χλ by dλ(M)=∑σ∈Snχλ(σ)∏i=1nmiσ(i).For λ=(k,1n−k), the immanant d(k,1n−k)(M) is called the hook immanant and denoted by dk(M). The hook immanant polynomial of matrix M is defined as dk(xIn−M), where In is the n×n identity matrix. Let G and G→ be a graph and a digraph, respectively. Denote by D(G) and A(G) (resp. D(G→) and A(G→)) the degree matrix and adjacency matrix of G (resp. G→). In this paper, we establish several immanantal inequalities for linear combination matrix of the form βD(G)+γA(G), where β and γ are real numbers. Applying these inequalities, we give the solutions of two open problems on immanants proposed by Chan and Lam. Furthermore, we characterize two types of hook immanantal equalities for the linear combination matrices βD(G)+γA(G) and βD(G→)+γA(G→). As applications, we provide dimensionality reduction formulas for the hook immanantal polynomials and hook immanants of graph matrices. And the classic known results of previous researchers can be derived.