%0 Journal Article
%T The second immanant of some combinatorial matrices
%J Transactions on Combinatorics
%I University of Isfahan
%Z 2251-8657
%A Bapat, R. B.
%A Sivasubramanian, Sivaramakrishnan
%D 2015
%\ 06/01/2015
%V 4
%N 2
%P 23-35
%! The second immanant of some combinatorial matrices
%K Immanant
%K distance matrix
%K laplacian
%R 10.22108/toc.2015.6237
%X Let $A = (a_{i,j})_{1 \leq i,j \leq n}$ be an $n \times n$ matrix where $n \geq 2$. Let $\det 2(A)$, its second immanant be the immanant corresponding to the partition $\lambda_2 = 2,1^{n-2}$. Let $G$ be a connected graph with blocks $B_1, B_2,\ldots, B_p$ and with $q$-exponential distance matrix $ED_G$. We give an explicit formula for $\det 2(ED_G)$ which shows that $\det 2(ED_G)$ is independent of the manner in which $G$'s blocks are connected. Our result is similar in form to the result of Graham, Hoffman and Hosoya and in spirit to that of Bapat, Lal and Pati who show that $\det ED_T$ where $T$ is a tree is independent of the structure of $T$ and only dependent on its number of vertices. Our result extends more generally to a product distance matrix associated to a connected graph $G$. Similar results are shown for the $q$-analogue of $T$'s laplacian and a suitably defined matrix for arbitrary connected graphs.
%U https://toc.ui.ac.ir/article_6237_0e3b4c61593d783ddddf34dff3214698.pdf