Graphs, friends and acquaintances

TL;DR

This paper discusses four fundamental graph theory results related to social relationships and their applications in various mathematical and network theories, highlighting their significance and ongoing research challenges.

Contribution

It provides commentary on four key graph theory results connected to social relations, illustrating their importance and current research directions.

Findings

The Handshake lemma's relation to graph colorings and Boolean algebra.

Application of a lemma on known and unknown people to Ramsey theory.

Theorem on friends in common related to distance-regularity and coding theory.

Abstract

As is well known, a graph is a mathematical object modeling the existence of a certain relation between pairs of elements of a given set. Therefore, it is not surprising that many of the first results concerning graphs made reference to relationships between people or groups of people. In this article, we comment on four results of this kind, which are related to various general theories on graphs and their applications: the Handshake lemma (related to graph colorings and Boolean algebra), a lemma on known and unknown people at a cocktail party (to Ramsey theory), a theorem on friends in common (to distance-regularity and coding theory), and Hall's Marriage theorem (to the theory of networks). These four areas of graph theory, often with problems which are easy to state but difficult to solve, are extensively developed and currently give rise to much research work. As examples of representative problems and results of these areas, which are discussed in this paper, we may cite the following: the Four Colors Theorem (4CTC), the Ramsey numbers, problems of the existence of distance-regular graphs and completely regular codes, and finally the study of topological proprieties of interconnection networks.