On the spectral distribution of large weighted random regular graphs

Bulk behavior of Schur–Hadamard products of symmetric random matrices

“…They found that when p = 0.5, the LSD is the familiar semi-circular law, but when p = 0.5, the limiting moments are polynomials in (2p − 1) whose coefficients could not be identified, and as p approaches 1 these moments approach the corresponding moments of the LSD of the Toeplitz matrix. Goldmakher et al [2013] considered randomly weighted sequences of d-regular graphs with size growing to ∞, which amounts to taking Schur-Hadamard product of random real symmetric weight matrices with the adjacency matrices of the graphs, and established the existence of a limiting spectral distribution that depends only on d and the distribution of the weights, under the usual decay condition on the number of k-cycles relative to the graph-size, for each k 3 (in the unweighted case, the limiting spectral distribution is the well-known Kesten's measure).…”