The second immanant of some combinatorial matrices

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.