Given a structured random matrix ensemble where each random variable occurs o(N) times in each row and the limiting rescaled spectral measure e μ exists, we fix a p 2 [1/2, 1] and study the ensemble of signed structured matrices by multiplying the (i, j)th and (j, i)th entries of a matrix by a randomly chosen ✏ij 2 {1, 1}, with Prob(✏ij = 1) = p (i.e., the Hadamard product). For p = 1/2 the lim...

Using Grassmann variables and an analogy with two dimensional electrostatics, we obtain the average eigenvalue distribution ρ(ω) of ensembles of N × N asymmetrically diluted Hopfield matrices in the limit N → ∞. We found that in the limit of strong dilution the distribution is uniform in a circle in the complex plane. Present address: Universidade Federal de Braśılia, Brazil Present address: Fa...

This paper presents some bounds on the number of Laplacian eigenvalues contained in various subintervals of [0, n] by using the matching number and edge covering number for G, and asserts that for a connected graph the Laplacian eigenvalue 1 appears with certain multiplicity. Furthermore, as an application of our result (Theorem 13), Grone and Merris’ conjecture [The Laplacian spectrum of graph...

We consider a class of matrices of the form Cn = (1/N)A 1/2 n XnBnX ∗ nA 1/2 n , where Xn is an n × N matrix consisting of i.i.d. standardized complex entries, A n is a non-negative definite Hermitian square-root of the non-negative definite matrix An, and Bn is diagonal with nonnegative diagonal entries. Under the assumption that the distribution of the eigenvalues of An and Bn converge to pro...

In this paper we study the distribution of eigenvalues of regular graphs , regular hypergraphs , and biregular bipartite graphs of given girth by considering the polynomials orthogonal with respect to the measures attached to the spectra of such graphs and to the continuous spectra of their ‘universal covers’ . Our estimates are tight for Biggs graphs and generalized polygons . We also give an ...