Hook immanantal inequalities for Laplacians of trees Original Research Article

For an irreducible character Xλ of the symmetric group Sn, indexed by the partition λ, the immanant function dλ, acting on an n × n matrix A = (aij), is defined as dλ(A) = Σσ Sn Xλ(σ)Πni = 1 aiσ(i). The associated normalized immanant is defined as where identity is the identity permutation. P. Heyfron has shown that for the partitions (k, 1n−k), the normalized immanant satisfies for all positive semidefinite Hermitian matrices A. When A is restricted to the Laplacian matrices of graphs, improvements on the inequalities above may be expected. Indeed, in a recent survey paper, R. Merris conjectured that wherever A is the Laplacian matrix of a tree. In this note, we establish a refinement for the family of inequalities in (1) when A is the Laplacian matrix of a tree, that includes (2) as a special case. These inequalities are sharp and equality holds if and only if A is the Laplacian matrix of the star. This is proved via the inequalities for k = 2, 3, …, n − 1, where A is the Laplacian matrix of a tree.