The second immanant of some combinatorial matrices

Document Type: Research Paper

Authors

1 Stat-Math Unit, ISI Delhi

2 Dept 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.

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