The coefficients of the immanantal polynomial

An expression of the coefficient of immanantal polynomial of an n × n matrix is present. Moreover, we give expressions of the coefficient of immanantal polynomials of combinatorial matrices (adjacency matrix, Laplacian matrix, signless Laplacian matrix). As applications, we show that the immanantal polynomials for Laplacian matrix and signless Laplacian matrix of bipartite graphs are the same. This is a generalization of the characteristic polynomial for Laplacian matrix and signless Laplacian matrix of bipartite graphs. Furthermore, we consider the relations between the characteristic polynomial and the immanantal polynomial for trees.