In this paper, we investigate the spectral properties of the adjacency and the Laplacian matrices of random graphs. We prove that: (i) the law of large numbers for the spectral norms and the largest eigenvalues of the adjacency and the Laplacian matrices; (ii) under some further independent conditions, the normalized largest eigenvalues of the Laplacian matrices are dense in a compact interval ...

In this paper, we investigate the spectral properties of the adjacency and the Laplacian matrices of random graphs. We prove that (i) the law of large numbers for the spectral norms and the largest eigenvalues of the adjacency and the Laplacian matrices; (ii) under some further independent conditions, the normalized largest eigenvalues of the Laplacian matrices are dense in a compact interval a...

We establish bounds on the spectral radii for a large class of sparse random matrices, which includes the adjacency matrices of inhomogeneous Erdős-Rényi graphs. For the Erdős-Rényi graph G(n, d/n), our results imply that the smallest and second-largest eigenvalues of the adjacency matrix converge to the edges of the support of the asymptotic eigenvalue distribution provided that d log n. Toget...

Let d≥3 be a fixed integer, and let A the adjacency matrix of random d-regular directed or undirected graph on n vertices. We show that there exists constant d>0 such P(Ais singular inR)≤n−d, for sufficiently large. This answers an open problem by Frieze Vu. The key idea is to study singularity probability over finite field Fp. proof combines local central limit theorem large deviation estimate.

Forty years ago, Goethals and Seidel showed that if the adjacency algebra of a strongly regular graph X contains a Hadamard matrix then X is of Latin square type or of negative Latin square type [8]. We extend their result to complex Hadamard matrices and find only three additional families of parameters for which the strongly regular graphs have complex Hadamard matrices in their adjacency alg...

Absfruct-This paper discusses an approximate solution to the weighted graph matching prohlem (WGMP) for both undirected and directed graphs. The W G M P is the problem of f inding the opt imum matching between two weighted graphs, which are graphs with weights at each arc. The proposed method employs an analytic, instead of a combinatorial or iterative, approach to the opt imum matching problem...

Tutte proved that, if two graphs, both with more than two vertices, have the same collection of vertex-deleted subgraphs, then the determinants of the two corresponding adjacency matrices are the same. In this paper, we give a geometric proof of Tutte’s theorem using vectors and angles. We further study the lowest eigenspaces of these adjacency matrices.