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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

Article 3, Volume 4, Issue 2, June 2015, Page 23-35  XML PDF (271 K)
Document Type: Research Paper
DOI: 10.22108/toc.2015.6237
Authors
R. B. Bapat1; Sivaramakrishnan Sivasubramanian 2
1Stat-Math Unit, ISI Delhi
2Dept of Mathematics, IIT Bombay
Abstract
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.
Keywords
Immanant; distance matrix; laplacian
Main Subjects
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.); 15A15 Determinants, permanents, other special matrix functions
References
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[12] S. Sivasubramanian, A q-analogue of Graham, Hoffman and Hosoyas theorem, Electron. J. Combin., 17 (2010) pp. 9.

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