On a generalization of the friendship theorem

Abstract

The Friendship Theorem states that if any two people, of a group of at least three people, have exactly one friend in common, then there is always a person who is everybody's friend. In this paper, we generalize the Friendship Theorem to the case that in a group of at least three people, if every two friends have one or two common friends and every pair of strangers have exactly one friend then there exist one person who is friend to everybody in the group. In particular, we show that the graph corresponding to this problem is of type G = K1 . (sK2 + tK3), where s and t are non-negative integers and Km is the complete graph on m vertices.