Kirchhoff Indexes of a network

In this work we define the effective resistance between any pair of vertices with respect to a value λ⩾0 and a weight ω on the vertex set. This allows us to consider a generalization of the Kirchhoff Index of a finite network. It turns out that λ is the lowest eigenvalue of a suitable semi–definite positive Schrödinger operator and ω is the associated eigenfunction. We obtain the relation between the effective resistance, and hence between the Kirchhoff Index, with respect to λ and ω and the eigenvalues of the associated Schrödinger operator. However, our main aim in this work is to get explicit expressions of the above parameters in terms of equilibrium measures of the network. From these expressions, we derive a full generalization of Foster’s formulae that incorporate a positive probability of remaining in each vertex in every step of a random walk. Finally, we compute the effective resistances and the generalized Kirchhoff Index with respect to a λ and ω for some families of networks with symmetries, specifically for weighted wagon–wheels and circular ladders.