University of IsfahanTransactions on Combinatorics2251-86574220150601The second immanant of some combinatorial matrices2335623710.22108/toc.2015.6237ENR. B. BapatStat-Math Unit, ISI DelhiSivaramakrishnan SivasubramanianDept of Mathematics, IIT BombayJournal Article20140220Let $A = (a_{i,j})_{1 leq i,j leq n}$ 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.http://toc.ui.ac.ir/article_6237_0e3b4c61593d783ddddf34dff3214698.pdf