Immanantal Polynomials of Laplacian Matrix of Trees

The immanant d () associated with the irreducible character of the symmetric group S n , indexed by the partition of n, acting on an nn matrix A = a ij ] is deened by d (A) = X 2Sn () n Y i=1 a ii(i) : For a tree T on n vertices, let L(T) denote its Laplacian matrix. Let x be an indeterminate variable and I be the n n identity matrix. The immanantal polynomial of T corresponding to d is deened as d (xI ? L(T)) = n X k=0 (?1) k c ;k(T) x n?k : The coeecients c ;k (T) admit various algebraic and topological interpretations for the tree T. We study the properties of c ;k (T) as well as upper and lower bounds on c ;k (T) and in particular show that c ;k (S(n)) c ;k (T) c ;k (P (n)) for all partitions and 0 k n, where S(n) and P(n) denote the star and the path on n vertices respectively. This answers some questions posed in 16] in the aarmative. We study the properties of c ;k (T) in more detail when is restricted to the family of hook immanants.