The Maximum Number of 3- and 4-Cliques within a Planar Maximally Filtered Graph

TL;DR

This paper proves the conjecture that the maximum number of 3- and 4-cliques in a Planar Maximally Filtered Graph (PMFG) with n vertices is 3n - 8 and n - 4, respectively, confirming previous heuristic claims.

Contribution

It provides a rigorous proof confirming the maximum number of 3- and 4-cliques in PMFGs, settling a recent conjecture in the field.

Findings

Maximum 3-cliques in PMFG is 3n - 8

Maximum 4-cliques in PMFG is n - 4

Confirmed heuristic conjectures with formal proof

Abstract

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 n vertices is 3n - 8 and n - 4 respectively. In this paper, we prove that this is indeed the case.