The emerging area of network science studies the structural characteristics of networks and dynamic processes on networks such as spread of epidemics and vulnerability of power grids to cascading failures etc. Treating each element of a graph as a resistance, Kirchhoff index defined by the chemistry community is sum of the effective resistances across all pairs of nodes of the graph. This index has been studied using the graph Laplacian matrix which is the same as the indefinite admittance matrix of a resistance network. In this paper we introduce the concept of Weighted Kirchhoff index of a graph and its relationship to Foster's theorems. We present a generalization of Foster's theorems that retains the circuit theoretic flavor and elegance of these theorems. Furthermore we also present a dual form of Foster's first theorem.
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