This paper focuses on the problem of finding clusterings with maximum modularity, a quality measure for graph partitions. Modularity has gained significant attention in various disciplines, particularly in complex systems, but its properties are not well understood. The authors prove the hardness of maximizing modularity in general graphs and under the restriction to cuts, providing an Integer Linear Programming (ILP) formulation. They also analyze the performance of the greedy agglomeration approach, showing that it can yield an approximation factor of no better than two in certain graph families. The paper includes theoretical foundations, hardness proofs, and insights into the behavior of the greedy algorithm, along with characterizations of optimal clusterings for specific graph structures.This paper focuses on the problem of finding clusterings with maximum modularity, a quality measure for graph partitions. Modularity has gained significant attention in various disciplines, particularly in complex systems, but its properties are not well understood. The authors prove the hardness of maximizing modularity in general graphs and under the restriction to cuts, providing an Integer Linear Programming (ILP) formulation. They also analyze the performance of the greedy agglomeration approach, showing that it can yield an approximation factor of no better than two in certain graph families. The paper includes theoretical foundations, hardness proofs, and insights into the behavior of the greedy algorithm, along with characterizations of optimal clusterings for specific graph structures.