Spectral graph theory is the study of properties of the Laplacian matrix or adjacency matrix associated with a graph. In this paper, we focus on the connection between the eigenvalues of the Laplacian matrix and graph connectivity. Also, we use the adjacency matrix of a graph to count the number of simple paths of length up to 3.

A graph can be associated with a matrix in several ways. For instance, by associating the vertices of the graph to the rows/columns and then using 1 to indicate an edge and 0 otherwise we get the adjacency matrix A. The combinatorial Laplacian matrix is defined by L = D − A where D is a diagonal matrix with diagonal entries the degrees and A is again the adjacency matrix. Both of these matrices...

The problem of relating the eigenvalues of the normalized Laplacian for a weighted graph G and G − H, for H a subgraph of G is considered. It is shown that these eigenvalues interlace and that the tightness of the interlacing is dependent on the number of nonisolated vertices of H. Weak coverings of a weighted graph are also defined and interlacing results for the normalized Laplacian for such ...

The problem of relating the eigenvalues of the normalized Laplacian for a weighted graph G and G − H, for H a subgraph of G is considered. It is shown that these eigenvalues interlace and that the tightness of the interlacing is dependent on the number of nonisolated vertices of H. Weak coverings of a weighted graph are also defined and interlacing results for the normalized Laplacian for such ...

Control of vehicle formations has emerged as a topic of significant interest to the controls community. In this paper, we merge tools from graph theory and control theory to derive stability criteria for vehicle formations. The interconnection between vehicles (i.e., which vehicles are sensed by other vehicles) is modeled as a graph, and the eigenvalues of the Laplacian matrix of the graph are ...

There are several reasons why the study of the spectrum of the Laplacian in a narrow neighborhood of an embedded graph is interesting. The graph can be embedded into a Euclidean space or it can be embedded into a manifold. In his pioneering work [3], Colin de Verdière used Riemannian metrics concentrated in a small neighborhood of a graph to prove that for every manifold M of dimension greater ...

A connected graph is called Q-controllable if its signless Laplacian eigenvalues are mutually distinct and main. Two graphs G and H are said to be Q-cospectral if they share the same signless Laplacian spectrum. In this paper, infinite families of Q-controllable graphs are constructed, by using the operator of rooted product introduced by Godsil and McKay. In the process, infinitely many non-is...

1 Laplacian Methods: An Overview 2 1.1 De…nition: The Laplacian operator of a Graph . . . . . . . . . . 2 1.2 Properties of the Laplacian and its Spectrum . . . . . . . . . . . 4 1.2.1 Spectrum of L and e L: Graph eigenvalues and eigenvectors: 4 1.2.2 Other interesting / useful properties of the normalized Laplacian (Chung): . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Laplacians of Weight...

In this work we consider the nonlinear graph p-Laplacian and the set of eigenvalues and associated eigenvectors of this operator defined by a variational principle. We prove a unifying nodal domain theorem for the graph p-Laplacian for any p ≥ 1. While for p > 1 the bounds on the number of weak and strong nodal domains are the same as for the linear graph Laplacian (p = 2), the behavior changes...