The Second Immanantal Polynomial for the Signless Laplacian Matrix of a Graph

The second immanantal polynomial is one of the important directions in algebraic theory. Let M=[mij] be an n×n matrix. The second immanant of matrix M is defined as d2(M)=∑σ∈Snχ(σ)∏i=1nmiσ(i), where χ is the irreducible character of the symmetric group Sn of degree n, corresponding to the partition (21,1n−2). Let G be a graph with n vertices. Denote by Q(G) the signless Laplacian matrix of G. The second signless Laplacian immanantal polynomial of G is defined as d2(xI−Q(G))=∑k=0n(−1)kck(G)xn−k, where ck(G) is the coefficient of this polynomial. This paper investigates fundamental properties of this polynomial. First, we give combinatorial expressions for the first few coefficients of the second signless Laplacian immanantal polynomial. Next, we prove that the polynomial has no zero or negative real roots for connected graphs. Furthermore, we show that there is an equivalence relation among three polynomials for regular graphs, which implies that if two regular graphs share the same characteristic polynomial, then they also share the same second signless Laplacian immanantal polynomial. Finally, we prove that paths and almost complete graphs are determined by their second signless Laplacian immanantal polynomials.