Effective resistances and Kirchhoff index in subdivision networks

Abstract

In this work we compute the effective resistances and the Kirchhoff Index of subdivision networks in terms of the corresponding parameters of the original network. Our techniques are based on the study of discrete operators using discrete Potential Theory. Starting from a given network G = ( V, E, c ), we add a new vertex v xy at every edge { x, y } ¿ E and define new conductances c ( x, v xy ) so as to satisfy the electrical compatibility condition 1 c ( x, y ) = 1 c ( x, v xy ) + 1 c ( y, v xy ) , in order to obtain a subdivision network G S = ( V S , E S , c ) . In this setting we prove how the solution to a compatible Poisson problem on the subdivision network G S can be related with the solution of a suitable Poisson problem on the inicial network G . We use contraction and extension of functions from C ( V S ) to C ( V ) and viceversa, respectively to achieve our result. As effective resistances are computed with the aid of a solution (no matter what) to a particular Poisson problem, we can stablish an affine relationship between effective resistances defined for vertices in G S and the effective resistances defined for vertices in G . Here the coefficients are simply weighted averages of the conductances previously stated. Finally we relate the Kirchhoff index of the subdivision network K (G S ) with the Kirchhoff index of the initial network K (G) by using the so called Green‘s kernel function because the desired parameter can be computed as the trace of the kernel.