Let $L(G)$ be the Laplacian matrix of a digraph $G$ and $S_k(G)$ sum $k$ largest absolute values eigenvalues $G$. $C_n^+$ with $n+1$ vertices obtained from directed cycle $C_n$ by attaching pendant arc whose tail is on $C_n$. A $\mathbb{C}_n^+$-free if it contains no $C_{\ell}^+$ as subdigraph for any $2\leq \ell \leq n-1$. In this paper, we present lower bounds $S_n(G)$ digraphs order $n$. We ...

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...

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...

Let G = (V,E) be a graph with vertex set V = {1, . . . , n} and edge set E. Throughout the paper, a graph G is undirected and simple (i.e., has no multi-edges or loops). We allow G to be disconnected. The Laplacian of G is the matrix L(G) = D−A, where D is the diagonal matrix whose entries are the degree of the vertices and A is the adjacency matrix of G. Chung’s normalized Laplacian L̃(G) [6] i...

Let Gl be the graph obtained from Kl by adhering the root of isomorphic trees T to every vertex of Kl, and dk−j+1 be the degree of vertices in the level j. In this paper we study the spectrum of the adjacency matrix A(Gl) and the Laplacian matrix L(Gl) for all positive integer l, and give some results about the spectrum of the adjacency matrix A(Gl) and the Laplacian matrix L(Gl). By using thes...