Calculating effective resistances on underlying networks of association schemes

Abstract

Recently, in the work of Jafarizadeh et al. [J. Phys, A: Math. Theor. 40, 4949 (2007); e-print arXiv:0705.2480], calculation of effective resistances on distance-regular networks was investigated, where in the first paper, the calculation was based on stratification and Stieltjes functions associated with the network, whereas in the latter one a recursive formula for effective resistances was given based on the Christoffel–Darboux identity. In this paper, evaluation of effective resistances on more general networks that are underlying networks of association schemes is considered, where by using the algebraic combinatoric structures of association schemes such as stratification and Bose–Mesner algebras, an explicit formula for effective resistances on these networks is given in terms of the parameters of the corresponding association schemes. Moreover, we show that for particular underlying networks of association schemes with diameter d such that the adjacency matrix A possesses d+1 distinct eigenvalues, all of the other adjacency matrices Ai, i≠0, 1 can be written as polynomials of A, i.e., Ai=Pi(A), where Pi is not necessarily of degree i. Then, we use this property for these particular networks and assume that all of the conductances except for one of them, say, c≡c1=1, are zero to give a procedure for evfor a galuating effective resistances on these networks. The preference of this procedure is that one can evaluate effective resistances by using the structure of their Bose–Mesner algebra without any need to know the spectrum of the adjacency matrices.