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.

New spectral graph theoretic conditions for synchronization in directed complex networks

Abstract: This paper proposes lower bounds for the coupling strengths of oscillators in directed networks to guarantee global synchronization. The novel idea of graph comparison from spectral graph theory is employed so that the topological features of a given network can be fully utilized to simplify computations. For large networks that can be decomposed into a set of smaller strongly connected components, the comparison can be carried out at the local level as well.

A system/graph theoretical analysis of attractor coders

Abstract: This paper provides links between the young field of attractor coding and the well-established fields of systems theory and graph theory. Attractor decoders are modeled as linear systems whose stability is both necessary and sufficient for convergence of the decoder. This stability is dictated by the location of the eigenvalues of the sparse state transition matrix of the system. The relationship between these eigenvalues, spatial causality of the system, and the patterns of interdependency between signal elements (or image pixels) is investigated for several cases using concepts from graph and matrix theory.

Spectral graph theory efficiently characterises ventilation heterogeneity in lung airway networks.

Abstract: This paper introduces a linear operator for the purposes of quantifying the spectral properties of transport within resistive trees, such as airflow in lung airway networks. The operator, which we call the Maury matrix, acts only on the terminal nodes of the tree and is equivalent to the adjacency matrix of a complete graph summarising the relationships between all pairs of terminal nodes. We show that the eigenmodes of the Maury operator have a direct physical interpretation as the relaxation, or resistive, modes of the network. We apply these findings to both idealised and image-based models of ventilation in lung airway trees and show that the spectral properties of the Maury matrix characterise the flow asymmetry in these networks more concisely than the Laplacian modes, and that eigenvector centrality in the Maury spectrum is closely related to the phenomenon of ventilation heterogeneity caused by airway narrowing or obstruction. This method has applications in dimensionality reduction in simulations of lung mechanics, as well as for characterisation of models of the airway tree derived from medical images.

Role of Adjacency Matrix in Graph Theory

Abstract: Graph theory is an applied branch of the mathematics which deals the problems, with the help of graphs. There are many applications of graph theory to a wide variety of subjects which include Operations Research, Physics, Chemistry, Economics, Genetics, Sociology, Computer Science, Engineering, Mechanical Engineering and the other branches of science. A graph can be represented inside a computer by using the adjacency matrix. We define adjacency matrix and observed based on adjacency matrix. We prove theorem of adjacency matrix and give example. Also draw a graph of adjacency matrix. We derive algorithm to found new adjacency matrix after fusion and fusion for connectedness. Also use fusion algorithm to check the connectedness.