Encyclopedia of Social Network Analysis and Mining

2018 Edition
| Editors: Reda Alhajj, Jon Rokne

Spectral Analysis

  • Xiao-Dong Zhang
Reference work entry
DOI: https://doi.org/10.1007/978-1-4939-7131-2_168



Network (graph)

A network G is a triple consisting of a node set V (G), a link set E(G), and a relation that associates each link with two nodes

Adjacency matrix

Let G = (V (G), E(G)) be a network with V (G) = {v1, ···, vn}.

The adjacency matrix A(G) = (aij) of G is n × n matrix with aij = 1 if vi is adjacent to vj, and 0 otherwise

Eigenvalues of a graph

All eigenvalues of the adjacency matrix A(G) of a graph G are called eigenvalues of G and denoted by

λ1 ≥ λ2 ≥ … ≥ λn

Degree diagonal matrix

The degree diagonal matrix D(G) of a network G is the diagonal matrix whose diagonal entries are degrees of the corresponding nodes

Laplacian matrix

The Laplacian matrix L(G) is defined be L(G) = D(G) − A(G), where D(G) is the degree diagonal matrix and A(G) is the adjacency matrix

Laplacian eigenvalues of a graph

All eigenvalues of the Laplacian matrix L(G) of a graph Gare called the Laplacian...

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This work is supported by the National Natural Science Foundation of China (Nos. 11531001 and 11271256), the Joint NSFC-ISF Research Program (jointly funded by the National Natural Science Foundation of China and the Israel Science Foundation (No. 11561141001), Innovation Program of Shanghai Municipal Education Commission (No. 14ZZ016), and Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130073110075).


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Authors and Affiliations

  1. 1.School of Mathematical Science, Ministry of Education Key Laboratory of Scientific and Engineering ComputingShanghai Jiao Tong UniversityShanghaiChina

Section editors and affiliations

  • Suheil Khoury
    • 1
  1. 1.American University of SharjahSharjahUnited Arab Emirates