ArticlePDF Available

The Friendship Theorem

Authors:
THE FRIENDSHIP THEOREM
Craig Huneke
The ‘friendship theorem’ can be stated as follows ([1, p. 183]):
Suppose in a group of at least three people we have the situation that any pair of persons
have precisely one common friend. Then there is always a person who is everybody’s friend.
The ﬁrst proof of this theorem was due to Paul Erd˝os, Alfred R´enyi, and Vera S´os [3].
Translating the theorem into graph theory yields the following theorem:
Theorem. If Gis a graph in which any two distinct vertices have exactly one common
neighbor, then Ghas a vertex joined to all others.
As a consequence, such graphs are completely determined; they consist of edge-disjoint
triangles around a common vertex. The best known and simplest proof is based on com-
puting the eigenvalues (and their multiplicities) of the square of the adjacency matrix of
the graph. Such a proof is given in [1]. In [6], a similar idea is used, but phrased in terms
of showing the graph has the structure of a ‘projective plane’. (See also [2].) In his recent
book [5 p. 466], West writes, “It is startling that such a combinatorial-sounding result
seems to have no short combinatorial proof. There do exist proofs avoiding eigenvalues (see
Hammersley, [4]), but they require complicated numerical arguments to eliminate regular
graphs.” The goal of this note is to provide one proof which is more combinatorial, and
another proof which does not explicitly use eigenvalues (though it does use traces), and in
some sense combines the combinatorics with the linear algebra.
I ﬁrst heard of this problem from William Lang as a graduate student in 1975; he chal-
lenged me to solve it. I used the idea of counting walks of length p, where pis a carefully
chosen prime number, to construct a proof that same year. I recently showed this proof to
my colleague Fred Galvin, who then signiﬁcantly simpliﬁed it; this resulted in our ﬁrst proof
following. The ﬁrst three paragraphs of the proof are standard; they reduce the problem to
a regular graph. However, we make this reduction in a slightly diﬀerent way than in the
references to this paper.
Proof of Theorem. We ﬁrst claim that if x, y Gand are not adjacent, then they have
the same degree (i.e., the same number of adjacent vertices). Let N(x) denote the vertices
adjacent to x(the ‘neighborhood’ of x). Deﬁne a map α:N(x)N(y) by sending a vertex
zN(x) to the common neighbor of zand y. This common neighbor cannot be x, as x
and yare not adjacent. The map is one-to-one since zN(x) is uniquely determined as
the common neighbor of xand α(z). By symmetry the map is onto.
Typeset by A
M
S-T
E
X
1
2 CRAIG HUNEKE
Suppose there is a vertex of degree k > 1. We claim that all vertices have degree k, unless
there is a universal friend. Let Abe the set of all vertices of degree k, let Bbe the set of all
vertices of degree diﬀerent from k, and assume for a contradiction that Bis nonempty. By
the ﬁrst claim, every vertex in Ais adjacent to every vertex in B. If Aor Bis a singleton,
then that singleton is a universal friend; otherwise, there are two diﬀerent vertices in A, and
they have two common neighbors in B, contradicting the hypothesis. It follows that Gis
k-regular, i.e., the degree of every vertex is k.
Next we claim that the number nof vertices in Gis exactly k(k1) + 1. This follows by
counting the paths of length two in G: by assumption there are n
2such paths. For each
vertex v, there are exactly k
2paths of length two having vin the middle, giving n·k
2total
paths of length two. Equating both counts permits us to conclude that n=k(k1) + 1.
We can assume that k3, else n= 3 and the theorem is clear.
Awalk of length non Gis an ordered sequence v0v1...vnof vertices such that viand vi+1
are neighbors. We say the walk is closed if vn=v0. A closed walk is considered to have a
starting point and an orientation, always returning to the starting vertex; thus, if uvwu is
a closed walk in G, then uvwu,vwuv,vuwv, etc. are considered distinct closed walks. It
follows that, if pis a prime number, then the number of closed walks of length pis divisible
by p.
For a ﬁxed vertex v, let f(n) be the number of walks from vto vof length n. If n > 1,
then the number of closed n-walks v0...vn2vn1vnfrom v=v0=vnwith vn2=vis
kf (n2), and the number of such walks with vn26=vis kn2f(n2). The total
number of walks v0v1....vn2from a ﬁxed vertex v=v0is kn2as Gis k-regular. Thus
f(n)=(k1)f(n2) + kn2. Let pbe a prime divisor of k1; then f(p)1 (mod
p). Finally, the total number of closed walks of length pis [k(k1) + 1]f(p)1 (mod p),
contradicting the fact that the number of such walks is divisible by p.
Although the preceding proof requires no linear algebra, one can compare this proof to
the proof of [1], which uses eigenvalues. Thinking about it gives another simple proof,
provided one knows a little linear algebra in characteristic p.
Second Proof of the Friendship Theorem. We again reduce to the case in which the graph is
k-regular, i.e., each vertex has exactly kadjacent vertices and the total number of vertices
is n=k(k1) + 1, with k3. Let G={v1, ..., vn}. We let Abe the adjacency matrix
of G, whose (i, j) entry is 1 if viand vjare adjacent, and 0 otherwise. The matrix Ahas
zeroes on the diagonal, so the trace of Ais 0. We let Bbe the nby nmatrix having a 1 in
every entry. The trace of Bis n.
By assumption and the fact that Gis k-regular, A2= (k1)I+B, and AB =kB,
where Iis the identity matrix of size nby n. We now pass to the ﬁeld Zp, where pis a
prime dividing k1. We continue to call the matrices Aand B, though we now think of
them with entries in Zp. Observe that both nand kare now equal to 1. Hence A2=B,
and furthermore AB =kB =B. It follows that for all l2, Al=B. Let trC denote the
trace of a square matrix C. In characteristic p,trAp= (trA)p. We reach a contradiction:
1 = n=trB =trAp= (trA)p= 0.
The relation between the ﬁrst proof and the second is simply that the trace of Apcounts
THE FRIENDSHIP THEOREM 3
the closed walks of length pin the graph. The relationship between the second proof and
the usual proof is clear: in characteristic 0, one computes the eigenvalues of A2and then
proves that Acould not have trace 0. The second proof takes advantage of the fact that
(a+b)p=ap+bpin characteristic p. This allows us to avoid the actual computation of the
eigenvalues, to push the calculation of the trace out to the pth power.
Bibliography
[1] M. Aigner and G. Ziegler, Proofs from the Book, Springer-Verlag, Berlin, 1999.
[2] J. Brunat, Una demostraci´o del teorema de l’amistat per metodes elementals, Butllet´
i Societat
Catalana de Matematiques 7(1992), pp. 75–80.
[3] P. Erd˝os, A. R´enyi, and V. S´os, On a problem of graph theory, Studia Sci. Math. 1(1966), pp.
215–235.
[4] J. Hammersley, The friendship theorem and the love problem, in Surveys in Combinatorics, London
Math. Soc. Lec. Notes 82 (1983), Cambridge Univ. Press, Cambridge, pp. 31–54.
[5] D. B. West, Introduction to Graph Theory, 2nd edition, Prentice Hall, Upper Saddle River, NJ,
2001.
[6] H. Wilf, The friendship theorem, Combinatorial Mathematics and its Applications, Proc. Conf.
Oxford, 1969 (1971), Academic Press, London and New York, pp. 307–309.
University Of Kansas, Lawrence, KS 66045, USA
... (Reference : The article by Craig Huneke [6]) ...
... Trivial.Main Proof of Friendship Theorem :Proof. (Reference : The article by Craig Huneke[6]) For a fixed vertex v, let f (n) be the number of walks from v to v of length n. If n > 1, then the number of closed n-walks ...
Preprint
Full-text available
Proof of Friendship Theorem
... Are there cases in which changes in are known a priori, without a previous knowledge of ? The " reversal " of Rényi entropy is easily explainable in topological terms, as a straightforward extension of the pointbased Rényi Friendship Theorem (Huneke, 2002). Indeed, the above mentioned correlation between Rényi entropy and thermodynamic free-energy can be elucidated via the Friendship Theorem, in terms of the vertices of a particular graph (Havrda and Charvat, 1967, Rényi, 1966), illustrated in Figure 1A. ...
... Indeed, the above mentioned correlation between Rényi entropy and thermodynamic free-energy can be elucidated via the Friendship Theorem, in terms of the vertices of a particular graph (Havrda and Charvat, 1967, Rényi, 1966), illustrated in Figure 1A. It is Huneke's simplified version of the Friendship Theorem (Huneke, 2002) that we give next. Friendship Theorem. ...
Preprint
Our paper is currently under review. If you want to quote it, please write: Tozzi A, Peters JF, Çankaya MN, Korbel J, Zare M, Papo D. 2016. Energetic Link Between Spike Frequencies and Brain Fractal Dimensions. viXra:1609.0105. Oscillations in brain activity exhibit a power law distribution which appears as a straight line when plotted on logarithmic scales in a log power versus log frequency plot. The line’s slope is given by a single constant, the power law exponent. Since a variation in slope may occur during different functional states, the brain waves are said to be multifractal, i.e., characterized by a spectrum of multiple possible exponents. A role for such non-stationary scaling properties has scarcely been taken into account. Here we show that changes in fractal slopes and oscillation frequencies, and in particular in electric spikes, are correlated. Taking into account techniques for parameter distribution estimates, which provide a foundation for the proposed approach, we show that modifications in power law exponents are associated with variations in the Rényi entropy, a generalization of Shannon informational entropy. Changes in Rényi entropy, in turn, are able to modify brain oscillation frequencies. Therefore, results point out that multifractal systems lead to different probability outcomes of brain activity, based solely on increases or decreases of the fractal exponents. Such approach may offer new insights in the characterization of neuroimaging diagnostic techniques and the forces required for transcranial stimulation, where doubts still exist as to the parameters that best characterize waveforms. SIGNIFICANCE STATEMENT The generalized informational entropy called “Rényi entropy” does not select the most appropriate probabilistic parameter as Shannon’s, rather it builds diversity profiles. By offering a continuum of possible diversity measures at many spatiotemporal levels, it is very useful in the evaluation of the fractal scaling occurring in the brain. Rényi entropy elucidates how power laws behaviours in cortical oscillations are able to modify electric spike frequencies. Through its links with the scale-free behavior of cortical fluctuations, Rényi entropy suggests that the brain changes its fractal exponents in order to control free-energy and scale the entropy of different functional states.
... Indeed, in →1limit we recover the Boltzmann distribution. The " reversal " of Rényi entropy is also explainable as a straightforward extension of the point-based Rényi Friendship Theorem (Huneke, 2002). The correlation between Rényi entropy and thermodynamic free energy can elucidated via the Friendship Theorem, in terms of the vertices of a particular graph (Havrda-Charvat, 1967, Rényi, 1966). ...
... The correlation between Rényi entropy and thermodynamic free energy can elucidated via the Friendship Theorem, in terms of the vertices of a particular graph (Havrda-Charvat, 1967, Rényi, 1966). It is C. Huneke's simplified version of the Friendship Theorem (Huneke, 2002) that we give next. ...
Preprint
A slightly different version of this manuscript has been published. Please quote as: Tozzi A, Peters JF, Cankaya MN. 2018. The informational entropy endowed in cortical oscillations. Cognitive Neurodynamics, 12(5), 501-507. DOI: 10.1007/s11571-018-9491-3. ------------------------- The brain electric activity exhibits a power law distribution which appears as a straight line when plotted on logarithmic scales in a log power versus log frequency plot. The line’s slope is given by a single constant, the power law exponent. Since a variation in slope may occur during different functional states, the brain currents are said to be multifractal, i.e. characterized by a spectrum of multiple possible exponents. A role for such non-stationary scaling properties in neural coding has scarcely been taken into account. Here we show that changes in fractal slopes and in the spike frequency are correlated. Taking into account two novel techniques for parameters distribution’s estimates, which provide a foundation for the proposed approach, we illustrate that modifications in power law exponents are associated with variations in the Rényi entropy, which is a generalization of Shannon informational entropy. Changes in Rényi entropy, in turn, are able to modify brain spike frequencies. Our results point up that multifractal systems lead to different probability outcomes of brain activity, based solely on increases or decreases of the fractal exponents. We offer new insights in the characterization of neuroimaging diagnostic techniques and the forces required for transcranial stimulation, where doubts still exist about the parameters of the waveforms to use. We anticipate our findings to be a starting point for testing psychological correlates of the neural activity under currents equipped with different power law slopes.
... c) If P is abelian then G ∩ P ∩ Z(N G (P )) = 1. The proposition below, a special case of the well known "friendship problem" in graph theory, will be needed in proving the case p = 3 for both of Theorem A and Theorem B. For the sake of completeness we present here a proof of this fact, see [3]. Proposition 2.2. ...
Article
We give character free proofs of two solvability theorems due to Isaacs.
... Such a graph is sometimes called a 'windmill graph'. Subsequently, many different proofs of the Friendship Theorem have been given [4,5,11]. ...
Article
Full-text available
For $r \ge 2$, an $r$-uniform hypergraph is called a friendship $r$-hypergraph if every set $R$ of $r$ vertices has a unique 'friend' - that is, there exists a unique vertex $x \notin R$ with the property that for each subset $A \subseteq R$ of size $r-1$, the set $A \cup \{x\}$ is a hyperedge. We show that for $r \geq 3$, the number of hyperedges in a friendship $r$-hypergraph is at least $\frac{r+1}{r} \binom{n-1}{r-1}$, and we characterise those hypergraphs which achieve this bound. This generalises a result given by Li and van Rees in the case when $r = 3$. We also obtain a new upper bound on the number of hyperedges in a friendship $r$-hypergraph, which improves on a known bound given by Li, van Rees, Seo and Singhi when $r=3$.
... In the same year, West gave a proof similar to that in [3], counting common neighbors and cycles [7]. Finally, Huneke gave in 2002 two proofs, one being more combinatorial and one that combines combinatorics and linear algebra [8]. ...
Article
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.
Chapter
A Clifford graph algebra GA(G) is a useful structure for studying a simple graph G with n vertices. Such an algebra associates each of its n generators with one of the n vertices of G in a way that depicts the connectivity of G in that any two generators anti-commute or commute depending on whether their corresponding vertices share or do not share an edge. We will construct the Clifford graph algebra for any windmill graph W(r, m), which consist of m copies of the complete graph $$K_r$$ adjoined at one common vertex; and for any Dutch windmill graph $$D^m_r$$ which consists of m copies of the r-cycle graph $$C_r$$ adjoined at one common vertex, then apply this algebraic theory to the class of 3-cycle graphs $$F_m=D^m_3$$ known as friendship graphs. Specifically, we will use the algebra $$GA\big (F_m\big )$$ to give a new proof of the fact that those simple graphs which posses the friendship property are precisely the friendship graphs.KeywordsClifford algebraWindmill graphDutch windmill graphFriendship graphMathematics Subject Classification (2010)Primary 15A66
Preprint
We analyze the design of mechanisms to rank individuals in communities where individuals only have local, ordinal information on the characteristics of their neighbors. In completely informative communities, we show that the planner can construct an ex-post incentive compatible and ex-post efficient mechanism if and only if all pair of individuals are observed by a third individual---every pair of individuals in the social network has a common friend. We use this insight to characterize the sparsest social network for which a complete ranking exists as the "friendship network" of Erd\H{o}s, R\'enyi, and S\'os (1966). When the social network is not completely informative, we show that any self-report which is not supported by a third party must be discarded. We provide two sufficient conditions on the social network under which an ex-post incentive compatible and ex-post efficient mechanism may be constructed: when the social network is bipartite or only formed of triangles. We use data on social networks from India and Indonesia to illustrate the results of the theoretical analysis. We measure information provided by the social network as the share of unique comparisons which can be obtained by friend-based comparisons (the density of the comparison network) and show that information varies greatly even for given density, across triangle comparisons are important at low densities, and information is close to an upper bound when degree is capped at a small value relative to the community size.
Article
A simple undirected graph G has the me2-property if every pair of distinct vertices of G is connected by a path of length 2, but this property does not survive the deletion of an edge. This article considers graphs that can be embedded as induced subgraphs of me2-graphs by the introduction of additional mutually non-adjacent vertices and suitably chosen edges. The main results concern such embeddings of trees.
Article
Can a Christian escape from a lion? How quickly can a rumor spread? Can you fool an airline into accepting oversize baggage? Recreational mathematics is full of frivolous questions where the mathematician's art can be brought to bear. But play often has a purpose. In mathematics, it can sharpen skills, provide amusement, or simply surprise, and books of problems have been the stock-in-trade of mathematicians for centuries. This collection is designed to be sipped from, rather than consumed in one sitting. The questions range in difficulty: the most challenging offer a glimpse of deep results that engage mathematicians today; even the easiest prompt readers to think about mathematics. All come with solutions, many with hints, and most with illustrations. Whether you are an expert, or a beginner or an amateur mathematician, this book will delight for a lifetime.
Proofs from the Book Una demostració del teorema de l'amistat per m etodes elementals
• M Aigner
• G Ziegler
M. Aigner and G. Ziegler, Proofs from the Book, Springer-Verlag, Berlin, 1999. [2] J. Brunat, Una demostració del teorema de l'amistat per m etodes elementals, ButlletíButlletí Societat Catalana de Matemàtiques 7 (1992), pp. 75–80.