  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.
