After proper rescaling and under some technical assumptions, the smallest eigenvalue of a sample covariance matrix with aspect ratio bounded away from 1 converges to the Tracy–Widom distribution. This complements the results on the largest eigenvalue, due to Soshnikov and Péché.

After proper rescaling and under some technical assumptions, the smallest eigenvalue of a sample covariance matrix with aspect ratio bounded away from 1 converges to the Tracy–Widom distribution. This complements the results on the largest eigenvalue, due to Soshnikov and Péché.

We prove that the distribution function of the largest eigenvalue in the Gaussian Unitary Ensemble (GUE) in the edge scaling limit is expressible in terms of Painlevé II. Our goal is to concentrate on this important example of the connection between random matrix theory and integrable systems, and in so doing to introduce the newcomer to the subject as a whole. We also give sketches of the resu...

The largest eigenvalue of the adjacency matrix of networks is a key quantity determining several important dynamical processes on complex networks. Based on this fact, we present a quantitative, objective characterization of the dynamical importance of network nodes and links in terms of their effect on the largest eigenvalue. We show how our characterization of the dynamical importance of node...

Nested split and double nested graphs (commonly named nested graphs) are considered. General statements regarding the signless Laplacian spectra are proven, and the nested graphs whose second largest signless Laplacian eigenvalue is bounded by a fixed integral constant are studied. Some sufficient conditions are provided and a procedure for classifying such graphs in particular cases is provide...

Among all connected cactuses with n vertices we find a unique graph whose largest eigenvalue (index, for short) is maximal.

Let G be a simple connected graph with pendant vertex set ∂V and nonpendant vertex set V0. The signless Laplacian matrix of G is denoted by Q(G). The signless Dirichlet eigenvalue is a real number λ such that there exists a function f ̸= 0 on V (G) such that Q(G)f(u) = λf(u) for u ∈ V0 and f(u) = 0 for u ∈ ∂V . The signless Dirichlet spectral radius λ(G) is the largest signless Dirichlet eigenva...

We show lower bounds for the smallest non-trivial eigenvalue, and smallest real portion of an eigenvalue, of the Laplacian of a non-reversible Markov chain in terms of an Evolving set quantity. A myriad of Cheeger-like inequalities follow for non-reversible chains, which even in the reversible case sharpen previously known results. The same argument also produces a new Cheeger-like inequality f...

This paper establishes converses to the well-known result: for any vector ũ such that the sine of the angle sin θ(u, ũ) = O( ), we have ρ(ũ) def = ũ∗Aũ ũ∗ũ = λ+O( ), where λ is an eigenvalue and u is the corresponding eigenvector of a Hermitian matrix A, and “∗” denotes complex conjugate transpose. It shows that if ρ(ũ) is close to A’s largest eigenvalue, then ũ is close to the corresponding ei...

We derive some basic results on the geometry of semidefinite programming (SDP) and eigenvalue-optimization , i.e., the minimization of the sum of the k largest eigenvalues of a smooth matrix-valued function. We provide upper bounds on the rank of extreme matrices in SDPs, and the first theoretically solid explanation of a phenomenon of intrinsic interest in eigenvalue-optimization. In the spect...