The second immanant of some combinatorial matrices

Let A = (a_{i,j})_{۱ \leq i,j \leq n} be an n \times n matrix‎ ‎where n \geq ۲‎. ‎Let \det ۲(A)‎, ‎its second immanant be the immanant‎ ‎corresponding to the partition \lambda_۲ = ۲,۱^{n-۲}‎. ‎Let G be a connected graph with blocks B_۱‎, ‎B_۲,\ldots‎, ‎B_p and with‎ ‎q-exponential distance matrix ED_G‎. ‎We give an explicit‎ ‎formula for \det ۲(ED_G) which shows that \det ۲(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