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 $dt(A)$, 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 $ED_G$...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
University of Isfahan
2015-06-01
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| Series: | Transactions on Combinatorics |
| Subjects: | |
| Online Access: | http://www.combinatorics.ir/pdf_6237_0e3b4c61593d783ddddf34dff3214698.html |
| Summary: | Let $A = (a_{i,j})_{1 leq i,j leq n}$ be an $n times n$ matrix
where $n geq 2$. 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 explicit
formula 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 $T$'s laplacian
and a suitably defined matrix for arbitrary connected graphs. |
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| ISSN: | 2251-8657 2251-8665 |