# Calculating effective resistances on underlying networks of association schemes - Mathematical Physics

Abstract: Recently, in Refs. \cite{jsj} and \cite{res2}, calculation of effectiveresistances on distance-regular networks was investigated, where in the firstpaper, the calculation was based on stratification and Stieltjes functionassociated with the network, whereas in the latter one a recursive formula foreffective resistances was given based on the Christoffel-Darboux identity. Inthis paper, evaluation of effective resistances on more general networks whichare underlying networks of association schemes is considered, where by usingthe algebraic combinatoric structures of association schemes such asstratification and Bose-Mesner algebras, an explicit formula for effectiveresistances on these networks is given in terms of the parameters ofcorresponding association schemes. Moreover, we show that for particularunderlying networks of association schemes with diameter $d$ such that theadjacency matrix $A$ possesses $d+1$ distinct eigenvalues, all of the otheradjacency matrices $A i$, $i eq 0,1$ can be written as polynomials of $A$,i.e., $A i=P iA$, where $P i$ is not necessarily of degree $i$. Then, we usethis property for these particular networks and assume that all of theconductances except for one of them, say $c\equiv c 1=1$, are zero to give aprocedure for evaluating effective resistances on these networks. Thepreference of this procedure is that one can evaluate effective resistances byusing the structure of their Bose-Mesner algebra without any need to know thespectrum of the adjacency matrices.