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Tuesday, November 13, 2018 - 15:35 to 16:35
SMS Seminar Hall
Dr. Mukesh Kumar Nagar
IIT Dharwad
Immanants of Laplacians of Trees
Abstract: Let $T$ be a tree on $n$ vertices with Laplacian matrix $L_T$ and $q$-Laplacian $\sL_{T}^{q}$. Let $\chi_k^{}$ be the character of the irreducible representation of $\SSS_n$ indexed by the hook partition $k,1^{n-k}$ and let $\od_k(L_T)$ be the normalized hook immanant of $L_T$ corresponding to the character $\chi_k^{}$. Inequalities are known for $\od_k(L_T)$ as $k$ increases. By assigning statistics to vertex orientations, we generalize these inequalities to the matrix $\sL_{T}^{q}$ for all $q \in \RR$ and to the bivariate $q,t$-Laplacian $\sL_{T}^{q,t}$ for a specific set of values $q,t \in \CC$. Our statistic based approach also gives generalizations of inequalities for the Hadamard-Marcus immanantal inequality changing index $k(L_T)$ of $L_T$ to the matrices $\sL_{T}^{q}$ and $\sL_{T}^{q,t}$ for several values of $q,t$. In this talk, we will discuss these generalized results of $\sL_{T}^q$ and $\sL_{T}^{q,t}$ . Csikv{\'a}ri defined a poset on the set of unlabelled trees on $n$ vertices that we denote as $\GTS_n$. %Let $T$ be a tree on $n$ vertices with Laplacian $L_T$. Among other results, he showed that going up on $\GTS_n$ has the following effect: \begin{enumerate} \item In absolute value, all coefficients of the characteristic polynomial of $L_T$ decrease. \item The algebraic connectivity of $T$ and the spectral radius of $L_T$ increase. \end{enumerate} In this talk, we will also discuss a generalization of these results to the immanantal polynomial of $\sL_{T}^{q}$. We will see that going up on $\GTS_n$ has the following effect: \begin{enumerate} \item In absolute value, all coefficients of all immanantal polynomials of $\sL_{T}^{q}$ decrease for all $q\in \RR$. \item The spectral radius of $\sL_{T}^{q}$ increases while the smallest eigenvalue of $\sL_{T}^{q}$ decreases for all $q\in \RR$. Further when $q\in \RR$ with $|q|\geq 1$, the second smallest eigenvalue of $\sL_{T}^{q}$ increases. \end{enumerate} These results give simple consequences about immanantal polynomials and eigenvalues to the $q,t$-Laplacians $\sL_T^{q,t}$ and exponential distance matrices $\ED_T$ and $\ED_T^{q,t}$ of a tree $T$.

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