Topic
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.
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01 Dec 2011
TL;DR: In this paper, a signless Laplacian for the distance matrix of a connected graph is introduced, called the distance signless lplacians, and the relation between the signless lemma and the distance spectra of a given connected graph G and a bipartite component in the complement G is established.
Abstract: We introduce a signless Laplacian for the distance matrix of a connected graph, called the distance signless Laplacian. We study the distance signless Laplacian spectrum of a connected graph. We show the equivalence between the distance signless Laplacian, distance Laplacian and the distance spectra for the class of transmission regular graphs. We also establish a relationship between the smallest eigenvalue of the distance signless Laplacian of a connected graph G and the existence of a bipartite component in the complement G.
8 citations
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TL;DR: The approach is to take well known techniques from finite dimensional matrix analysis and show how they can be generalized for graph Laplacians and investigate how the perturbation of the graph can affect the eigenvalues.
Abstract: This paper deals with spectral graph theory issues related to questions of monotonicity and comparison of eigenvalues. We consider finite directed graphs with non symmetric edge weights and we introduce a special self-adjoint operator as the sum of two non self-adjoint Laplacians. We investigate how the perturbation of the graph can affect the eigenvalues. Our approach is to take well known techniques from finite dimensional matrix analysis and show how they can be generalized for graph Laplacians.
8 citations
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TL;DR: In this paper, the quantum central limit theorem for the adjacency matrix of a growing regular graph in the vacuum and deformed vacuum states has been proved and the condition for the growth is described in terms of simple statistics arising from the stratification of the graph.
Abstract: We propose the quantum probabilistic techniques to obtain the asymptotic spectral distribution of the adjacency matrix of a growing regular graph. We prove the quantum central limit theorem for the adjacency matrix of a growing regular graph in the vacuum and deformed vacuum states. The condition for the growth is described in terms of simple statistics arising from the stratification of the graph. The asymptotic spectral distribution of the adjacency matrix is obtained from the classical reduction.
8 citations
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01 Jan 2016TL;DR: In this paper, the authors explore recent development on various problems related to graph indices in trees, focusing on indices based on distances between vertices, vertex degrees, or on counting vertex or edge subsets of different kinds.
Abstract: In this chapter we explore recent development on various problems related to graph indices in trees. We focus on indices based on distances between vertices, vertex degrees, or on counting vertex or edge subsets of different kinds. Some of the indices arise naturally in applications, e.g., in chemistry, statistical physics, bioinformatics, and other fields, and connections are also made to other branches of graph theory, such as spectral graph theory. We will be particularly interested in the extremal values (maxima and minima) for different families of trees and the corresponding extremal trees. Moreover, we review results for random trees, consider localized versions of different graph indices and the associated notions of centrality, and finally discuss inverse problems, where one wants to find trees for which a specific graph index has a prescribed value.
8 citations
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01 Jan 2012TL;DR: This chapter introduces the methods for defining the data similarity (or dissimilarity) and introduces the preliminary spectral graph theory to analyze the data geometry.
Abstract: In applications, a high-dimensional data is given as a discrete set in a Euclidean space. If the points of data are well sampled on a manifold, then the data geometry is inherited from the manifold. Since the underlying manifold is hidden, it is hard to know its geometry by the classical manifold calculus. The data graph is a useful tool to reveal the data geometry. To construct a data graph, we first find the neighborhood system on the data, which is determined by the similarity (or dissimilarity) among the data points. The similarity information of data usually is driven by the application in which the data are used. In this chapter, we introduce the methods for defining the data similarity (or dissimilarity). We also introduce the preliminary spectral graph theory to analyze the data geometry. In Section 1, the construction of neighborhood system on data is discussed. The neighborhood system on a data set defines a data graph, which can be considered as a discrete form of a manifold. In Section 2, we introduce the basic concepts of graphs. In Section 3, the spectral graph analysis is introduced as a tool for analyzing the data geometry. Particularly, the Laplacian on a graph is briefly discussed in this section. Most of the materials in Sections 2 and 3 are found in [1–3].
8 citations