Modularity of nearly complete graphs and bipartite graphs
Combinatorics
2023-12-21 v2 Social and Information Networks
Abstract
It is known that complete graphs and complete multipartite graphs have modularity zero. We show that the least number of edges we may delete from the complete graph to obtain a graph with non-zero modularity is . Similarly we determine the least number of edges we may delete from or add to a complete bipartite graph to reach non-zero modularity. We give some corresponding results for complete multipartite graphs, and a short proof that complete multipartite graphs have modularity zero. We also analyse the modularity of very dense random graphs, and in particular we find that there is a transition to modularity zero when the average degree of the complementary graph drops below 1.
Keywords
Cite
@article{arxiv.2311.06875,
title = {Modularity of nearly complete graphs and bipartite graphs},
author = {Colin McDiarmid and Fiona Skerman},
journal= {arXiv preprint arXiv:2311.06875},
year = {2023}
}