The expected eigenvalue distribution of a large regular graph

By Brendan D. McKay. 1981

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Citations in the corpus

Decentralized gradient methods: does topology matter? (2020, background): “These graphs are known to be Ramanujan graphs with high-probability, i.e., they have the smallest | λ 2 | among all graphs with the same degree (McKay, 1981).”
On a class of free Lévy laws related to a regression problem (2004, background): “) = δ0. The laws of free Meixner type correspond to various reparametrizations of (2) and occurred in many places in the literature, see [3], [13], [14], [18, Corollary 7.2], [22, Example 5.6], [26], [33], [40]. In particular, recursion (3) for b ≥ 0 is a reparametrization of ﬁve recursions that app”
Distribution of Eigenvalues of Real Symmetric Palindromic Toeplitz Matrices and Circulant Matrices (2005, background): “ for each N there are only ﬁnitely many N×N d-regular graphs, and we choose uniformly from this set. While d-regular graphs are a subset of real symmetric matrices, they have diﬀerent behavior. McKay [McK] proved the density of eigenvalues of d-regular graphs is given by Kesten’s Measure, not the se”
SPECTRAL DISTRIBUTIONS OF ADJACENCY AND LAPLACIAN MATRICES OF RANDOM GRAPHS (2010, background): “It is the limit of the empirical distributions of the eigenvalues of the random d-regular graphs, see [36].”
Poisson–Dirichlet distribution for random Belyi surfaces (2005, unknown): “We also mention an early result of McKay [37], who showed that spectral density of random k-regular graphs converges to Kesten’s measure, that is, a measure supported on [−2√k − 1,2√k − 1 ] and given by νk = k 2π √ 4(k − 1) − t2 k2 − t2 .”
The spectral gap of dense random regular graphs (2016, methodology): “Let us remark that the proof of the main statement of this sub-section is a rather standard application of the method of simple switchings introduced by Senior [25] and developed by McKay and Wormald (see [21] as well as survey [30]).”
Adjacency matrices of random digraphs: singularity and anti-concentration (2015, background): “Among others, we refer the reader to [2], [3], [12], [14], [26], and [35] for results dealing with the eigenvalue distribution.”
Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees (2013, background): “In particular, McKay showed in 1981 [14] that for every fixed p, the normalized pth moments of a sequence {Ad} of random m-regular graphs of increasing size d → ∞ satisfy:”
The measurable Kesten theorem (2011, background): “Theorem 4 is related to a paper of Serre [24] (see also [19]) that studies asymptotic properties of graph sequences.”
Around the circular law (2011, background): “The Hermitian (actually symmetric) version of this measure is known as the Kesten-McKay distribution for random non-oriented r-regular graphs, see [94, 110].”
Eigenvalue statistics for beta-ensembles (2003, background): “The study of random matrices emerged in the late 1920’s (with the publishing of Wishart’s most important work in 1928) and 1930’s (with Hsu’s work [41]), and they are a very quickly growing field of research, with communities like nuclear physics [23, 33, 94], multivariate statistics [14, 47, 70], a”
Sparse regular random graphs: Spectral density and eigenvectors (2009, methodology): “A famous such example is McKay’s derivation of the limiting empirical spectrum of random d-regular graphs on n vertices, as d is fixed and n grows to infinity [45].”
The Role of Network Topology for Distributed Machine Learning (2019, background): “With high probability, random d-regular graphs are expanders with γ(An) = d− 2 √ d− 1 [17].”