The (n-1)-th Laplacian Immanantal Polynomials of Graphs

Let χn−1(σ) denote the irreducible character of the symmetric group Sn corresponding to the partition (n−1,1). For an n×n matrix M=(mi,j), we denote its (n−1)-th immanant by dn−1(M). Let G be a simple connected graph and let L(G) and Q(G) denote the Laplacian matrix and the signless Laplacian matrix of G, respectively. The (n−1)-th Laplacian (respectively, signless Laplacian) immanantal polynomial of G is defined as dn−1(xI−L(G)) (respectively, dn−1(xI−Q(G))). In this paper, we partially resolve Chan’s open problem by establishing that the broom graph minimizes dn−1(L(T)) among all trees with given diameter. Furthermore, we give combinatorial expressions for the first five coefficients of the (n−1)-th Laplacian immanantal polynomial dn−1(xI−L(G)). We also investigate the characterizing properties of this polynomial and present several graphs that are uniquely determined by it. Additionally, for the (n−1)-th signless Laplacian immanantal polynomial dn−1(xI−Q(G)), we show that the multiplicity of root 1 is bounded below by the star degree of G.