Eigenvalue monotonicity of q-Laplacians of trees along a poset

Let T be a tree on n vertices with q-Laplacian matrix LTq. Let GTSn be the generalized tree shift poset on the set of unlabelled trees with n vertices. We prove that for all q∈R, going up on GTSn has the following effect: the spectral radius of LTq increases while the smallest eigenvalue of LTq decreases. We also prove that for all q∈R with |q|≥1, going up on GTSn increases the second smallest eigenvalue of LTq. These generalize known results for eigenvalues of the Laplacian matrix of trees. As a corollary, we obtain consequences about the eigenvalues of q,t-Laplacians and exponential distance matrices of trees.