# On the spectral distribution of large weighted random regular graphs

@article{Goldmakher2013OnTS,
title={On the spectral distribution of large weighted random regular graphs},
author={Leo Goldmakher and Cap Khoury and Steven J. Miller and Kesinee Ninsuwan},
journal={arXiv: Probability},
year={2013}
}
• Published 28 June 2013
• Mathematics
• arXiv: Probability
McKay proved that the limiting spectral measures of the ensembles of $d$-regular graphs with $N$ vertices converge to Kesten's measure as $N\to\infty$. In this paper we explore the case of weighted graphs. More precisely, given a large $d$-regular graph we assign random weights, drawn from some distribution $\mathcal{W}$, to its edges. We study the relationship between $\mathcal{W}$ and the associated limiting spectral distribution obtained by averaging over the weighted graphs. Among other…
7 Citations

## Figures from this paper

Spectral distributions of periodic random matrix ensembles
Koloğlu, Kopp and Miller compute the limiting spectral distribution of a certain class of real random matrix ensembles, known as [Formula: see text]-block circulant ensembles, and discover that it is
The limiting spectral measure for an ensemble of generalized checkerboard matrices
• Mathematics
• 2020
Random matrix theory successfully models many systems, from the energy levels of heavy nuclei to zeros of $L$-functions. While most ensembles studied have continuous spectral distribution, Burkhardt
Distribution of Eigenvalues of Weighted, Structured Matrix Ensembles
• Mathematics
Integers
• 2015
It is proved that the limiting signed rescaled spectral measure is the semi-circle and a closed-form expression for the expected value is derived and the asymptotics for the variance for the number of vertices in at least one crossing are determined.
Random matrix ensembles with split limiting behavior
• Mathematics
Random Matrices: Theory and Applications
• 2018
We introduce a new family of [Formula: see text] random real symmetric matrix ensembles, the [Formula: see text]-checkerboard matrices, whose limiting spectral measure has two components which can be
Spectral statistics of non-Hermitian random matrix ensembles
• Mathematics
Random Matrices: Theory and Applications
• 2019
Recently Burkhardt et al. introduced the [Formula: see text]-checkerboard random matrix ensembles, which have a split limiting behavior of the eigenvalues (in the limit all but [Formula: see text] of
1 0 A pr 2 01 8 SPECTRAL STATISTICS OF NON-HERMITIAN RANDOM MATRIX ENSEMBLES
• Mathematics
• 2018
Recently Burkhardt et. al. introduced the k-checkerboard random matrix ensembles, which have a split limiting behavior of the eigenvalues (in the limit all but k of the eigenvalues are on the order
Bulk behaviour of Schur-Hadamard products of symmetric random matrices
• Mathematics
• 2014
We develop a general method for establishing the existence of the Limiting Spectral Distributions (LSD) of Schur–Hadamard products of independent symmetric patterned random matrices. We apply this

## References

SHOWING 1-10 OF 46 REFERENCES
Eigenvalue distribution of large weighted bipartite random graphs
We study eigenvalue distribution of the adjacency matrix $A^{(N,p, \alpha)}$ of weighted random bipartite graphs $\Gamma= \Gamma_{N,p}$. We assume that the graphs have $N$ vertices, the ratio of
The Distribution of the Largest Nontrivial Eigenvalues in Families of Random Regular Graphs
• Mathematics
Exp. Math.
• 2008
These graphs have important applications in communication network theory, allowing the construction of superconcentrators and nonblocking networks, as well as in coding theory and cryptography, and are well modeled by the β = 1 Tracy-Widom distribution for several families.
Connectivity for Random Graphs from a Weighted Bridge-Addable Class
The generalisation is to consider `weighted' random graphs, sampled from a suitable more general distribution, where the focus is on the bridges.
Eigenvalue distribution of large weighted random graphs
• Mathematics
• 2004
We study eigenvalue distribution of the adjacency matrix A(N,p) of weighted random graphs Γ=ΓN,p. We assume that the graphs have N vertices and the average number of edges attached to one vertex is
Random Graphs from a Weighted Minor-Closed Class
It is found that earlier results extend naturally in both directions, to general well-behaved classes of graphs, and to the weighted framework, for example results concerning the probability of a random graph being connected; and the 2-core which are new even for the uniform (unweighted) case.
From the Littlewood-Offord problem to the Circular Law: universality of the spectral distribution of random matrices
• Mathematics
• 2008
The famous \emph{circular law} asserts that if $M_n$ is an $n \times n$ matrix with iid complex entries of mean zero and unit variance, then the empirical spectral distribution (ESD) of the
Approximating the statistics of various properties in randomly weighted graphs
• Mathematics, Computer Science
SODA '11
• 2011
This paper defines a family of weighted graph properties and shows that for each property in this family, the problem of computing the kth moment of the corresponding random variable admits a fully polynomial-time randomized approximation scheme (FPRAS) for every fixed k.
Spectral measure of large random Hankel, Markov and Toeplitz matrices
• Mathematics
• 2003
We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure. For Hankel and Toeplitz matrices generated by i.i.d. random variables {X k } of unit variance,
Wegner estimate and level repulsion for Wigner random matrices
• Mathematics
• 2008
We consider $N\times N$ Hermitian random matrices with independent identically distributed entries (Wigner matrices). The matrices are normalized so that the average spacing between consecutive
The Limiting Spectral Measure for Ensembles of Symmetric Block Circulant Matrices
• Mathematics, Computer Science
• 2010
It is proved that the limiting spectral distribution exists and is determined by the pattern of the independent elements within an m-period, depending not only on the frequency at which each element appears, but also on the way the elements are arranged.