is the diagonal matrix of vertex degrees of G and A(G) is the adjacency matrix ofG. The eigenvalues of L(G) are called the Laplacian eigenvalues and denoted by λ1 ≥ λ2 ≥ · · · ≥ λn = 0. It is well known that λ1 ≤ n. We denote the number of spanning trees (also known as complexity) of G by κ(G). The following formula in terms of the Laplacian eigenvalues of G is well known (see, for example, [2]...

In this paper, all connected graphs with the fourth largest Laplacian eigenvalue less than two are determined, which are used to characterize all connected graphs with exactly three Laplacian eigenvalues no less than two. Moreover, we determine bipartite graphs such that the adjacency matrices of their line graphs have exactly three nonnegative eigenvalues. © 2003 Elsevier Ltd. All rights reser...

Using Lotker’s interlacing theorem on the Laplacian eigenvalues of a graph in [5] and Wang and Belardo’s interlacing theorem on the signless Laplacian eigenvalues of a graph in [6], we in this note obtain spectral conditions for some Hamiltonian properties of graphs. 2010Mathematics Subject Classification : 05C50, 05C45

Let G be a simple graph with adjacency matrix A (= AG). The eigenvalues and the spectrum of A are also called the eigenvalues and the spectrum of G, respectively. If we consider a matrix Q = D + A instead of A, where D is the diagonal matrix of vertex–degrees (in G), we get the signless Laplacian eigenvalues and the signless Laplacian spectrum, respectively. For short, the signless Laplacian ei...

This paper presents a new method for estimating the eigenvalues of the Laplacian matrix associated with the graph describing the network topology of a multi-agent system. Given an approximate value of the average of the initial condition of the network state and some intermediate values of the network state when performing a Laplacian-based average consensus, the estimation of the Laplacian eig...

Abstract. A signed graph Γ = (G, σ) consists of an unsigned graph G = (V, E) and a mapping σ : E → {+,−}. Let Γ be a connected signed graph and L(Γ),L(Γ) be its Laplacian matrix and normalized Laplacian matrix, respectively. Suppose μ1 ≥ · · · ≥ μn−1 ≥ μn ≥ 0 and λ1 ≥ · · · ≥ λn−1 ≥ λn ≥ 0 are the Laplacian eigenvalues and the normalized Laplacian eigenvalues of Γ, respectively. In this paper, ...

The toughness $t(G)$ of a graph $G=(V,E)$ is defined as $t(G)=\min\{\frac{|S|}{c(G-S)}\}$, in which the minimum taken over all $S\subset V$ such that $G-S$ disconnected, where $c(G-S)$ denotes number components $G-S$. We present two tight lower bounds for terms Laplacian eigenvalues and provide strong support conjecture better bound which, if true, implies both bounds, improves generalizes know...

By introducing a weight function to the Laplace operator, Bakry and Émery defined the “drift Laplacian” to study diffusion processes. Our first main result is that, given a Bakry–Émery manifold, there is a naturally associated family of graphs whose eigenvalues converge to the eigenvalues of the drift Laplacian as the graphs collapse to the manifold. Applications of this result include a new re...