Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Spectral-based weighted undirected graph partitioning algorithm

Abstract: During the initial partitioning phase of multilevel method,this paper proposed the algorithm of spectral partitioning for weighted undirected graph(SPWUG) and gave the Fiedler vector calculation of Laplacian matrix using the Lanczos iteration method.The components of the Fiedler vector were weighted on the corresponding vertices of graph such that differences of the components provide information about the distances between the vertices.According to the distances,the SPWUG algorithm computed the initial partition of the coarsest graph by extending the Laplacian spectral graph theory from the unweighted undirected graph into the weighted undirected graph.The experimental evaluations show that the SPWUG algorithm produces the performance improvement.Meanwhile,the analysis shows that the approximate global minimum partition of the coarsest graph may be the local minimum partition of the finest graph and need to strengthen the global search ability of refinement algorithm in the refinement phase.

Block Matrix Representation of a Graph Manifold Linking Matrix Using Continued Fractions

Abstract: We consider the block matrices and 3-dimensional graph manifolds associated with a special type of tree graphs. We demonstrate that the linking matrices of these graph manifolds coincide with the reduced matrices obtained from the Laplacian block matrices by means of Gauss partial diagonalization procedure described explicitly by W. Neumann. The linking matrix is an important topological invariant of a graph manifold which is possible to interpret as a matrix of coupling constants of gauge interaction in Kaluza-Klein approach, where 3-dimensional graph manifold plays the role of internal space in topological 7-dimensional BF theory. The Gauss-Neumann method gives us a simple algorithm to calculate the linking matrices of graph manifolds and thus the coupling constants matrices.

Applications of rational difference equations to spectral graph theory

Abstract: We study a general class of recurrence relations that appear in the application of a matrix diagonalization procedure. We find a general closed formula and determine the analytical properties of th...

Energy and laplacian spectrum of c4c8(s) nanotori and nanotube

Abstract: The spectrum of a finite graph is by definition the spectrum of the adjacency matrix, that is, its set of eigenvalues together with their multiplicities. The sum of the absolutes of these eigenvalues is the energy of graph. The Laplace spectrum of a finite undirected graph without loops is the spectrum of the Laplace matrix. There are some topological indices related the Laplacian spectrum. In this paper, using a mathematical model for C4C8(S) that introduced in Ref.[26], we write a MATHEMATICA program to compute the energy and Laplacian spectrum of molecular graph of arbitrary C4C8(S) nanotori and nanotube.

Optimizing Pinning Control of Directed Networks Using Spectral Graph Theory

Abstract: Pinning control of a complex network aims at aligning the states of all the nodes to an external forcing signal by controlling a small number of nodes in the network. An algebraic graph-theoretic condition has been proposed to optimize pinning control for both undirected and directed network. The problem we are trying to solve in this paper is the optimization of pinning control in directed networks, where the effectiveness of pinning control can be measured by the smallest eigenvalue of the submatrix obtained by deleting the rows and columns corresponding to the pinned nodes from the symmetrized Laplacian matrix of the network when coupling strength and individual node dynamics are given. By analysing the spectral properties using the topology information of the directed network, a necessary condition that ensure the smallest eigenvalue greater than 0 is obtained. Upper bounds and lower bounds of the smallest eigenvalue are proved, and then an algorithm that optimize pinning scheme to improve the smallest eigenvalue is proposed. Illustrative examples are shown to demonstrate our theoretical results in the paper.