Authors: Majstorović, Snježana
Stevanović, Dragan 
Title: A note on graphs whose largest eigenvalue of the modularity matrix equals zero
Journal: Electronic Journal of Linear Algebra
Volume: 27
First page: 611
Last page: 618
Issue Date: 1-Jan-2014
Rank: M22
Informally, a community within a graph is a subgraph whose vertices are more connected to one another than to the vertices outside the community. One of the most popular community detection methods is the Newman’s spectral modularity maximization algorithm, which divides a graph into two communities based on the signs of the principal eigenvector of its modularity matrix in the case that the modularity matrix has positive largest eigenvalue. Newman defined a graph to be indivisible if its modularity matrix has no positive eigenvalues. It is shown here that a graph is indivisible if and only if it is a complete multipartite graph.
Keywords: Community structure | Complete multipartite graph | Largest eigenvalue | Modularity matrix
Publisher: International Linear Algebra Society
Project: Slovenian Research Agency, Projects P1-0285 and J1-4021
Graph theory and mathematical programming with applications in chemistry and computer science 

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