Bapat, R., Sivasubramanian, S. (2015). The second immanant of some combinatorial matrices. Transactions on Combinatorics, 4(2), 23-35. doi: 10.22108/toc.2015.6237

R. B. Bapat; Sivaramakrishnan Sivasubramanian. "The second immanant of some combinatorial matrices". Transactions on Combinatorics, 4, 2, 2015, 23-35. doi: 10.22108/toc.2015.6237

Bapat, R., Sivasubramanian, S. (2015). 'The second immanant of some combinatorial matrices', Transactions on Combinatorics, 4(2), pp. 23-35. doi: 10.22108/toc.2015.6237

Bapat, R., Sivasubramanian, S. The second immanant of some combinatorial matrices. Transactions on Combinatorics, 2015; 4(2): 23-35. doi: 10.22108/toc.2015.6237

The second immanant of some combinatorial matrices

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.

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