Effective resistance matrices of weighted threshold graphs

Abstract

A threshold graph is generated from a single node by repeatedly adding either a node $i$ connected to all existing nodes with a common link weight $w_i >0 $ or a node $i$ connected to none. Let $ G_w $ be a weighted threshold graph encoded by the weight vector $ w = (w_1, w_2, \ldots, w_N) $ with $w_i \geq 0$. A closed-form expression for the pseudoinverse of its Laplacian matrix $Q_w$ is derived via spectral decomposition, which yields an explicit formula for the effective resistance matrix $ \Omega_w $. We present a detailed structural characterization of the matrix $\Omega_w$ and determine a subset of the spectrum of the matrix $ \Omega_w $ in terms of the weights $w_i$. As an application, we show that when the missing links of a threshold graph are sequentially added in nondecreasing order of effective resistance, the threshold property of the graph is preserved at each step until the complete graph of the same size is obtained.