On asymptotic solvability of random graph's laplacians

We prove that for all values of the edge probability p(n) the largest eigenvalue of a random graph G(n,p) satisfies almost surely: \lambda_1(G)=(1+o(1))max{\sqrt{\Delta},np}, where \Delta is a maximal degree of G, and the o(1) term tends to zero as max{\sqrt{\Delta},np} tends to infinity.

We investigate the statistics of the largest eigenvalue, $\lambda_{\rm max}$, in an ensemble of $N\times N$ large ($N\gg 1$) sparse adjacency matrices, $A_N$. The most attention is paid to the distribution and typical fluctuations of $\lambda_{\rm max}$ in the vicinity of the percolation threshold, $p_c=\frac{1}{N}$. The overwhelming majority of subgraphs representing $A_N$ near $p_c$ are exponentially distributed linear subchains, for which the statistics of the normalized l...

We consider a class of sparse random matrices, which includes the adjacency matrix of Erd\H{o}s-R\'enyi graphs $\mathcal G(N,p)$ for $p \in [N^{\varepsilon-1},N^{-\varepsilon}]$. We identify the joint limiting distributions of the eigenvalues away from 0 and the spectral edges. Our result indicates that unlike Wigner matrices, the eigenvalues of sparse matrices satisfy central limit theorems with normalization $N\sqrt{p}$. In addition, the eigenvalues fluctuate simultaneously...

Eigenvectors of large matrices (and graphs) play an essential role in combinatorics and theoretical computer science. The goal of this survey is to provide an up-to-date account on properties of eigenvectors when the matrix (or graph) is random.

We propose a general approach to the description of spectra of complex networks. For the spectra of networks with uncorrelated vertices (and a local tree-like structure), exact equations are derived. These equations are generalized to the case of networks with correlations between neighboring vertices. The tail of the density of eigenvalues $\rho(\lambda)$ at large $|\lambda|$ is related to the behavior of the vertex degree distribution $P(k)$ at large $k$. In particular, as ...

We introduce a powerful analytic method to study the statistics of the number $\mathcal{N}_{\textbf{A}}(\gamma)$ of eigenvalues inside any contour $\gamma \in \mathbb{C}$ for infinitely large non-Hermitian random matrices ${\textbf A}$. Our generic approach can be applied to different random matrix ensembles, even when the analytic expression for the joint distribution of eigenvalues is not known. We illustrate the method on the adjacency matrices of weighted random graphs wi...

We discuss various aspects of the statistical formulation of the theory of random graphs, with emphasis on results obtained in a series of our recent publications.

Networks are often studied using the eigenvalues of their adjacency matrix, a powerful mathematical tool with a wide range of applications. Since in real systems the exact graph structure is not known, researchers resort to random graphs to obtain eigenvalue properties from known structural features. However, this theory is far from intuitive and often requires training of free probability, cavity methods or a strong familiarity with probability theory. In this note we offer ...

We give inequalities relating the eigenvalues of the adjacency matrix and the Laplacian of a graph, and its minimum and maximum degrees. The results are applied to derive new conditions for quasi-randomness of graphs.

The Laplacian matrix of a simple graph is the difference of the diagonal matrix of vertex degree and the (0,1) adjacency matrix. In the past decades, the Laplacian spectrum has received much more and more attention, since it has been applied to several fields, such as randomized algorithms, combinatorial optimization problems and machine learning. This paper is primarily a survey of various aspects of the eigenvalues of the Laplacian matrix of a graph for the past teens. In a...