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The second immanant of some combinatorial matrices | ||
Transactions on Combinatorics | ||
مقاله 3، دوره 4، شماره 2، شهریور 2015، صفحه 23-35 اصل مقاله (270.73 K) | ||
نوع مقاله: Research Paper | ||
شناسه دیجیتال (DOI): 10.22108/toc.2015.6237 | ||
نویسندگان | ||
R. B. Bapat1؛ Sivaramakrishnan Sivasubramanian* 2 | ||
1Stat-Math Unit, ISI Delhi | ||
2Dept of Mathematics, IIT Bombay | ||
چکیده | ||
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. | ||
کلیدواژهها | ||
Immanant؛ distance matrix؛ laplacian | ||
مراجع | ||
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