The maximum number of 3- and 4-cliques within a planar maximally filtered graph

•Embeddings of nn-vertex triangulations in the form of maximal planar graphs.•Applying the generating operations proposed by Eberhard to construct the maximal planar graphs.•Any maximal planar graph can be transformed to a standard spherical triangulation.•The standard spherical triangulation structure always contains the maximum number of 3- and 4-cliques.

Planar Maximally Filtered Graphs (PMFG) are an important tool for filtering the most relevant information from correlation based networks such as stock market networks. One of the main characteristics of a PMFG is the number of its 3- and 4-cliques. Recently in a few high impact papers it was stated that, based on heuristic evidence, the maximum number of 3- and 4-cliques that can exist in a PMFG with nn vertices is 3n−83n−8 and n−4n−4 respectively. In this paper, we prove that this is indeed the case.