When the largest eigenvalue of the modularity and normalized modularity matrix is zero

TL;DR

This paper proves that the only graphs with a zero largest eigenvalue in their modularity matrices are complete or complete multipartite graphs, confirming a conjecture and characterizing their spectral properties.

Contribution

It provides a complete characterization of graphs with zero largest eigenvalue in their modularity and normalized modularity matrices, confirming a longstanding conjecture.

Findings

Complete and complete multipartite graphs have zero as the largest eigenvalue of their modularity matrices.

The modularity and normalized modularity matrices of these graphs are negative semidefinite.

Only these graphs exhibit this spectral property among all simple graphs.

Abstract

In July 2012, at the Conference on Applications of Graph Spectra in Computer Science, Barcelona, D. Stevanovic posed the following open problem: which graphs have the zero as the largest eigenvalue of their modularity matrix? The conjecture was that only the complete and complete multipartite graphs. They indeed have this property, but are they the only ones? In this paper, we will give an affirmative answer to this question and prove a bit more: both the modularity and the normalized modularity matrix of a graph is negative semidefinite if and only if the graph is complete or complete multipartite.