In this paper graphs with the largest Laplacian eigenvalue at most 4 are characterized. Using this we show that the graphs with the largest Laplacian eigenvalue less than 4 are determined by their Laplacian spectra. Moreover, we prove that ones with no isolated vertex are determined by their adjacency spectra.

Many graphs arising in various information networks exhibit the “power law” behavior – the number of vertices of degree k is proportional to k−β for some positive β. We show that if β > 2.5, the largest eigenvalue of a random power law graph is almost surely (1 + o(1)) √ m where m is the maximum degree. Moreover, the k largest eigenvalues of a random power law graph with exponent β have power l...

We establish precise right-tail small deviation estimates for the largest eigenvalue of real symmetric and complex Hermitian matrices whose entries are independent random variables with uniformly bounded moments. The proof relies on a Green function comparison along continuous interpolating matrix flow long time. Less also obtained in left tail.

In 7] and 8], the power and Lanczos algorithms with random start for estimating the largest eigenvalue of an n n large symmetric positive deenite matrix were analyzed. In this paper we continue this study by estimating an eigenvector corresponding to the largest eigenvalue. We analyze polynomial algorithms using Krylov information for two error criteria: the randomized error and the randomized ...

In this paper, using variational analysis and optimization techniques, we examine some fundamental analytic properties of Z-eigenvalues of a real symmetric tensor with even order. We first establish that the maximum Z-eigenvalue function is a continuous and convex function on the symmetric tensor space, and so, provide formulas of the convex conjugate function and ε-subdifferential of the maxim...

In this paper, we give infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [−1− √ 2,−2) and also infinitely many examples of (non-isomorphic) connected k-regular graphs with smallest eigenvalue in half open interval [α1,−1− √ 2) where α1 is the smallest root(≈ −2.4812) of the polynomial x3 + 2x2 − 2x − 2. From these results, we ...