Eigenvalues of Adjacency and Laplacian Matrices of Bracelet—Kn Graph

Laplacian and the Adjacency Matrices

Proof. We first recall that every non-singular matrix B can be written B = QR, where Q is an orthonormal matrix Q and R is upper-triangular matrix R with positive diagonals1 We will use a slight variation of this fact, writing B = RQ. Now, since QT = Q−1, QAQT has exactly the same eigenvalues as A. Let Rt be the matrix t ∗R+ (1− t)I, and consider the family of matrices Mt = RtQAQR t , as t goes...

Graph Embeddings and Laplacian Eigenvalues

Graph embeddings are useful in bounding the smallest nontrivial eigenvalues of Laplacian matrices from below. For an n×n Laplacian, these embedding methods can be characterized as follows: The lower bound is based on a clique embedding into the underlying graph of the Laplacian. An embedding can be represented by a matrix Γ; the best possible bound based on this embedding is n/λmax(Γ Γ). Howeve...

Laplacian eigenvalues and optimality: II. The Laplacian of a graph

Spectral Distributions of Adjacency and Laplacian Matrices of Random Graphs

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...

Bounds for Laplacian Graph Eigenvalues

Let G be a connected simple graph whose Laplacian eigenvalues are 0 = μn (G) μn−1 (G) · · · μ1 (G) . In this paper, we establish some upper and lower bounds for the algebraic connectivity and the largest Laplacian eigenvalue of G . Mathematics subject classification (2010): 05C50, 15A18.