Friendship Theorem Research Articles

Interlace polynomials of friendship graphs

In this paper, we study the interlace polynomials of friendship graphs, that is, graphs that satisfy the Friendship Theorem given by Erdos, Renyi and Sos. Explicit formulas, special values and behavior of coefficients of these polynomials are provided. We also give the interlace polynomials of other similar graphs, such as, the butterfly graph.

The Friendship Theorem

Summary In this article we prove the friendship theorem according to the article [1], which states that if a group of people has the property that any pair of persons have exactly one common friend, then there is a universal friend, i.e. a person who is a friend of every other person in the group

On generalization of the friendship theorem

On generalization of the friendship theorem

On a question of Sós about 3‐uniform friendship hypergraphs

AbstractThe well‐known Friendship Theorem states that if G is a graph in which every pair of vertices has exactly one common neighbor, then G has a single vertex joined to all others (a “universal friend”). V. Sós defined an analogous friendship property for 3‐uniform hypergraphs, and gave a construction satisfying the friendship property that has a universal friend. We present new 3‐uniform hypergraphs on 8, 16, and 32 vertices that satisfy the friendship property without containing a universal friend. We also prove that if n ≤ 10 and n ≠ 8, then there are no friendship hypergraphs on n vertices without a universal friend. These results were obtained by computer search using integer programming. © 2008 Wiley Periodicals, Inc. J Combin Designs 16: 253–261, 2008

Extension of Strongly Regular Graphs

The Friendship Theorem states that if any two people in a party have exactly one common friend, then there exists a politician who is a friend of everybody. In this paper, we generalize the Friendship Theorem. Let $\lambda$ be any nonnegative integer and $\mu$ be any positive integer. Suppose each pair of friends have exactly $\lambda$ common friends and each pair of strangers have exactly $\mu$ common friends in a party. The corresponding graph is a generalization of strongly regular graphs obtained by relaxing the regularity property on vertex degrees. We prove that either everyone has exactly the same number of friends or there exists a politician who is a friend of everybody. As an immediate consequence, this implies a recent conjecture by Limaye et. al.

Infinite generalized friendship graphs

We give necessary and sufficient conditions for the existence of infinite generalized friendship graphs and show that there are 2° non-isomorphic ones of each admissible order c and chromatic number. Further we prove that such graphs and their complements are almost always regular of degree equal to the order and that various generalizations of the Friendship Theorem do not hold for infinite generalized friendship graphs.

Graphs in which each m-tuple of vertices is adjacent to the same number n of other vertices

Let G be a finite graph in which each m-tuple of mutually distinct vertices is adjacent to exactly n other vertices. If m ≥3 then G is isomorphic to the complete m+ n graph. For completeness we state the friendship theorem of Erdös, Rényi, and Sós and a theorem of Bose and Shrikhande, both of which deal with the case m=2.

A Variation of the Friendship Theorem

A friendship set is a finite set with a symmetric nonreflexive binary relation, called on, satisfying : (i) if a is not on b, then there exists a unique element which is on both a and b ; (ii) if a is on b, then there exists at most one element which is on both a and b. It is shown that for any nontrivial friendship set, there exists a least integer m, called the degree of the set, such that each element is on exactly m or $m - 1$ elements. If every element is on the same number of elements, the set is said to be homogeneous. The number of elements in a friendship set of degree m is either $m^2 + 1$ or $m^2 - m + 1$, depending on whether it is homogeneous or not. There are at most four distinct homogeneous friendship sets. Necessary conditions for the existence of nonhomogeneous friendship sets are given in terms of the irreducibility of certain polynomials.

The friendship theorem

The “Friendship Theorem” states that, in a party of n persons, if every pair of persons has exactly one common friend, then there is someone in the party who is everyone else's friend. (It is assumed that “friendship” is a symmetric, irreflexive relation). This theorem has been proved by Erdös, Rényi and Sós; by G. Higman, and by Wilf. Each of these proofs has relied, at some point, on sophisticated mathematics, and Wilf has stated that no wholly elementary proof is known. The purpose of this note is to give an elementary graph theoretic proof in which the key step is a simple counting argument due to Ball. The Friendship Theorem is then applied to obtain a recent result of Ryser on intersections of finite sets.