A poset denoted $\mathsf{GTS}_n$ on the set of unlabeled trees with $n$ vertices was defined by Csikvàri. He showed that several tree parameters are monotonic as one goes up this $\mathsf{GTS}_n$ poset. Let $T$ be a tree on $n$ vertices and let $\mathcal{L}_q^T$ be the $q$-analogue of its Laplacian. For all $q\in \mathbb{R}$, I will discuss monotonicity of the largest and the smallest eigenvalues of $\mathcal{L}_q^T$ along the $\mathsf{GTS}_n$ poset.

For a partition $\lambda \vdash n$, let the normalized immanant of $\mathcal{L}_q^T$ indexed by $\lambda$ be denoted as $\overline{\mathrm{Imm}}_{\lambda}(\mathcal{L}_q^T)$. Monotonicity of $\overline{\mathrm{Imm}}_{\lambda}(\mathcal{L}_q^T)$ will be discussed when we go up along $\mathsf{GTS}_n$ or when we change the size of the first row in the hook partition $(\lambda=k,1^{n-k})$ and the two row partition $\lambda=(n-k,k)$. We will also discuss monotocity of each coefficients in the $q$-Laplacian immanantal polynomials $\overline{\mathrm{Imm}}_{\lambda}(xI-\mathcal{L}_q^T)$ when we go up along $\mathsf{GTS}_n$. At the end of this talk, I will discuss our ongoing research projects and future plans.

This is a joint work with Prof. A. K. Lal (IITK) and Prof. S. Sivaramakrishnan (IITB).