The second immanant of some combinatorial matrices

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Let $A = (a_{i,j})_{1 leq i,j leq n}$ be an $n times n$ matrixwhere $n geq 2$. Let $dt(A)$, 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 given an explicitformula for $dt(ED_G)$ which shows that $dt(ED_G)$ is independent of the manner in which the 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 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 $T$ʹs laplacian and a suitably defined matrix for arbitrary connected graphs.