Hook immanantal and Hadamard inequalities for q-Laplacians of trees

Let T be a tree on n vertices with Laplacian matrix L and q-Laplacian Lq. Let χk be the character of the irreducible representation of Sn indexed by the hook partition k,1n−k and let d‾k(L) be the normalized hook immanant of L corresponding to the character χk. Inequalities for d‾k(L) as k increases are known. By assigning a statistic to vertex orientations on trees, we generalize these inequalities to the q-analogue Lq of L for all q∈R and to the bivariate q,t-Laplacian Lq,t for some values q, t. Our statistic based approach also generalizes several other inequalities including the changing index k(L) of the Hadamard inequality for L, to the matrix Lq and Lq,t. Thus, we extend several results about L to Lq which includes the case when Lq is not positive semidefinite.