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, ...

Among all trivalent graphs on n vertices, let Gn be one with the smallest possible eigenvalue gap. (The eigenvalue gap is the difference between the two largest eigenvalues of the adjacency matrix; for regular graphs, it equals the second smallest eigenvalue of the Laplacian matrix.) We show that Gn is unique for each n and has maximum diameter. This extends work of Guiduli and solves a conject...

Spectral Graph Theory is the study of the spectra of certain matrices defined from a given graph, including the adjacency matrix, the Laplacian matrix and other related matrices. Graph spectra have been studied extensively for more than fifty years. In the last fifteen years, interest has developed in the study of generalized Laplacian matrices of a graph, that is, real symmetric matrices with ...

In this lecture, I will discuss the adjacency matrix of a graph, and the meaning of its smallest eigenvalue. This corresponds to the largest eigenvalue of the Laplacian, which we will examine as well. We will relate these to bounds on the chromatic numbers of graphs and the sizes of independent sets of vertices in graphs. In particular, we will prove Hoffman’s bound, and some generalizations. W...

We developed a procedure of reducing the number of vertices and edges of a given tree, which we call the " tree simplification procedure, " without changing its topological information. Our motivation for developing this procedure was to reduce computational costs of graph Laplacian eigenvalues of such trees. When we applied this procedure to a set of trees representing dendritic structures of ...

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity ecc (G) of a graph G is the mean value of eccentricities of all vertices of G. The harmonic index H (G) of a graph G is defined as the sum of 2 di+dj over all edges vivj of G, where di denotes the degree of a vertex vi in G. In this paper, we determine the unique tree with minimum average...

In this note we generalize our results on synchronization of homogeneous Lur’e networks by static, relative state information based protocols from [23] to the case that for each agent only relative measurements are available. We establish sufficient conditions under which a dynamic synchronization protocol exists for such networks. These conditions involve feasibility of a two LMI’s together wi...

The spectral theory of higher-order symmetric tensors is an important tool for revealing some important properties of a hypergraph via its adjacency tensor, Laplacian tensor, and signless Laplacian tensor. Owing to the sparsity of these tensors, we propose an efficient approach to calculate products of these tensors and any vectors. By using the state-of-the-art L-BFGS approach, we develop a fi...

We consider the random Schrödinger operator−ε−2∆(d) +ξ (ε)(x), with ∆(d) the discrete Laplacian on Zd and ξ (ε)(x) are bounded and independent random variables, on sets of the form Dε := {x ∈ Zd : xε ∈ D} for D bounded, open and with a smooth boundary, and study the statistics of the Dirichlet eigenvalues in the limit ε ↓ 0. Assuming Eξ (ε)(x) = U(xε) holds for some bounded and continuous funct...

We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are presented in the context of a number of computer-generated conjectures.