The friendship theorem

The Friendship Theorem

In this paper we explore the friendship theorem, which in graph theory is stated as if we have a finite graph in which any two nodes have exactly one common neighbor, then Then there is a node which is adjacent to all other nodes. We provide a common proof of the friendship theorem, followed by two extensions. The first extension relates to the number of common neighbors a node must have. The s...

On generalization of the friendship theorem

According to the classical Friendship Theorem, if G is a finite simple graph such that each two distinct vertices have a unique common neighbor, then G has a dominating vertex. Jian Shen showed that if require only non-neighbors to have a unique common neighbor, then one of the following possibilities hold: i) G has dominating vertex, or ii) G is a strongly regular graph, or iii) G is a bi-regu...

The Königsberg Bridge Problem and the Friendship Theorem

This development provides a formalization of undirected graphs and simple graphs, which are based on Benedikt Nordhoff and Peter Lammich’s simple formalization of labelled directed graphs [4] in the archive. Then, with our formalization of graphs, we have shown both necessary and sufficient conditions for Eulerian trails and circuits [2] as well as the fact that the Königsberg Bridge problem do...

A Probe into the Concept of Friendship in the Qur’ān

The topic of human relationships—be they positive and sympathetic, or negative and hostile—receives a great amount of attention in the Qur’ān. Human beings need to be aware of both their friends and their enemies in order to learn how to engage with both groups, how to make additional friends and defend themselves from harm caused by enemies, and, if possible, turn these enemies into friends. T...

The Association between Friendship Companionship and Friendship Temporality: the Moderating Roles of Friendship Context and Ethnicity