The Friendship Theorem
نویسنده
چکیده
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 second relaxes the friendship condition such that any two nodes can have no common neighbor or one common neighbor.
منابع مشابه
The Friendship Problem on Graphs
In this paper we provide a purely combinatorial proof of the Friendship Theorem, which has been first proven by P. Erdös et al. by using also algebraic methods. Moreover, we generalize this theorem in a natural way, assuming that every pair of nodes occupies ≥ 2 common neighbors. We prove that every graph, which satisfies this generalized -friendship condition, is a regular graph.
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