HOOK IMMANANTAL INEQUALITIES FOR LAPLACIANS OF TREES

For an irreducible character chi(lambda) of the symmetric group S-n, i ndexed by the partition lambda, the immanant function d(lambda), actin g on an n x n matrix A = (a(ij)), is defined as d(lambda)(A) = Sigma(s igma is an element of Sn) chi(lambda)(sigma)Pi(i=1)(n)a(i sigma(i)). T he associated normalized immanant <(d)over bar(lambda)> is defined as <(d)over bar(lambda)> = d(lambda)/chi(lambda)(identity) where identity is the identity permutation. P. Heyfron has shown that for the partit ions (k, 1(n-k)), the normalized immanant <(d)over bar(k)> satisfies d et A = <(d)over bar(1)>(A) less than or equal to <(d)over bar(2)>(A) l ess than or equal to ... less than or equal to <(d)over bar(n)>(A) = p er A (1) for all positive semidefinite Hermitian matrices A. When A is restricted to the Laplacian matrices of graphs, improvements on the i nequalities above may be expected. Indeed, in a recent survey paper, R . Merris conjectured that <(d)over bar(n-1)>(A) less than or equal to n-2/n-1 <($)over bar(n)>(A) (2) whenever A is the Laplacian matrix of a tree. In this note, we establish a refinement for the family of ineq ualities in (1) when A is the Laplacian matrix of a tree, that include s (2) as a special case. These inequalities are sharp and equality hol ds if and only if A is the Laplacian matrix of the star. This is prove d via the inequalities <(d)over bar(k)>(A) - <(d)over bar(k-1)>(A) les s than or equal to <(d)over bar(k+1)>(A) - <(d)over bar(k)>(A) for k = 2,3,..., n - 1, where A is the Laplacian matrix of a tree. (C) Elsevi er Science Inc., 1997.