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 L-q. Let X-k be the character of the irreducible representation of S-n indexed by the hook partition k,1(n-k) and let (d) over bar (k)(L) be the normalized hook immanant of L corresponding to the character X-k. Inequalities for (d) over bar (k)(L) as k increases are known. By assigning a statistic to vertex orientations on trees, we generalize these inequalities to the q-analogue L-q of L for all q is an element of R and to the bivariate q, t-Laplacian L-q,L-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 L-q and L-q,L-t. Thus, we extend several results about L to L-q which includes the case when L-q is not positive semidefinite. (C) 2017 Elsevier Inc. All rights reserved.