Classification of polyhedral graphs by numbers of common neighbours

TL;DR

This paper introduces a classification scheme for polyhedral graphs based on the number of common neighbors between vertex pairs, linking graph theory with applications in networks and data science.

Contribution

It provides a comprehensive framework to classify polyhedra by neighbor counts, either characterizing all such graphs or constructing infinite families for each type.

Findings

Classified all polyhedra types based on common neighbor counts.

Constructed infinite families or proved non-existence for certain types.

Linked the classification to strongly regular and Deza graphs.

Abstract

We propose a classification of polyhedra (planar, $3$-connected graphs) according to their type i.e., their set of quantities of common neighbours for each pair of distinct vertices. For every (finite) set of non-negative integers, we either classify all the polyhedra of that type, or construct infinitely many polyhedra of that type, or prove that none exist. This problem is related to the theory of strongly regular and Deza graphs, distances in graphs, and degree sequences. There is potential for application to complex networks and data science.