the second immanant of some combinatorial matrices

Let a = (ai,j) 1≤i,j≤n be an n × n matrix where n ≥ 2. let det2(a), its second immanant be the immanant corresponding to the partition λ2 = 2, 1^n−2. let g be a connected graph with blocks b1, b2, . . . , bp and with q-exponential distance matrix edg. we give an explicit formula for det2(edg) which shows that det2(edg) 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 edt 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.