For a graph, the least signless Laplacian eigenvalue is the least eigenvalue of its signless Laplacian matrix. This paper investigates how the least signless Laplacian eigenvalue of a graph changes under some perturbations, and minimizes the least signless Laplacian eigenvalue among all the nonbipartite graphs with given matching number and edge cover number, respectively.

Spectral graph theory gives an algebraical approach to analyze the dynamics of a network by using the matrix that represents the network structure. However, it is not easy for social networks to apply the spectral graph theory because the matrix elements cannot be given exactly to represent the structure of a social network. The matrix element should be set on the basis of the relationship betw...

Let G be a graph on n vertices. Its Laplacian matrix is the n-by-n matrix L(G) = D(G) A(G), where A(G) is the familiar (0, 1) adjacency matrix, and D(G) is the diagonal matrix of vertex degrees. This is primarily an expository article surveying some of the many results known for Laplacian matrices. Its six sections are: Introduction, The Spectrum, The Algebraic Connectivity, Congruence and Equi...

We descibe a new spectral algorithm for reordering a sparse symmetric matrix to reduce its envelope size The ordering is computed by associating a Laplacian matrix with the given matrix and then sorting the components of a speci ed eigenvec tor of the Laplacian This Laplacian eigenvector solves a continuous relaxation of a related discrete problem called the minimum sum problem The permutation ...

The smallest eigenvalues and the associated eigenvectors (i.e.,eigenpairs) of a graph Laplacian matrix have been widelyused for spectral clustering and community detection. How-ever, in real-life applications the number of clusters or com-munities (say, K) is generally unknown a-priori. Conse-quently, the majority of the existing methods either chooseK heuristically or t...

We study the matrices Qk of in-forests of a weighted digraph Γ and their connections with the Laplacian matrix L of Γ. The (i, j) entry of Qk is the total weight of spanning converging forests (in-forests) with k arcs such that i belongs to a tree rooted at j. The forest matrices, Qk, can be calculated recursively and expressed by polynomials in the Laplacian matrix; they provide representation...

In this paper, we describe the use of graph-spectral techniques and their relationship to Riemannian geometry for the purposes of segmentation and grouping. We pose the problem of segmenting a set of tokens as that of partitioning the set of nodes in a graph whose edge weights are given by the geodesic distances between points in a manifold. To do this, we commence by explaining the relationshi...

Our goal is to use the properties of the adjacency/Laplacian matrix of graphs to first understand the structure of the graph and, based on these insights, to design efficient algorithms. The study of algebraic properties of graphs is called algebraic graph theory. One of the most useful algebraic properties of graphs are the eigenvalues (and eigenvectors) of the adjacency/Laplacian matrix.

• Capital letters represent matrices and bold lower-case letters represent vectors. For a matrix A, aij denotes the element in row i and column j; for the vector x, xi denotes the i th entry in the vector. • Various special matrices are represented by the following conventions: The adjacency matrix is denoted A; the degree matrix is denoted D; the Laplacian D − A is denoted L. The Laplacian can...