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... Since then, the importance of this theorem was recognized even in the field of block designs and coding theory as well as in the set theory. Accordingly, it has been actively studied and there have been several different approaches to providing a simpler proof (see [6,8,9,10,11]). This paper was motivated by the question: ...
The Friendship Theorem states that if in a party any pair of persons has precisely one common friend, then there is always a person who is everybody's friend and the theorem has been proved by Paul Erd\"{o}s, Alfr\'{e}d R\'{e}nyi, and Vera T. S\'{o}s in 1966. This paper was written in response to the question, ``What would happen if the hypothesis stating that any pair of persons has exactly one common friend were replaced with one stating that any pair of persons warms to exactly one person?". We call a digraph obtained in this way a friendship digraph. It is easy to check that a symmetric friendship digraph becomes a friendship graph if each directed cycle of length two is replaced with an edge. Based on this observation, one can say that a friendship digraph is a generalization of a friendship graph. In this paper, we provide a digraph formulation of the Friendship Theorem by defining friendship digraphs as those in which any two distinct vertices have precisely one common out-neighbor. We also establish a sufficient and necessary condition for the existence of friendship digraphs.
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 adjoined at one common vertex; and for any Dutch windmill graph which consists of m copies of the r-cycle graph adjoined at one common vertex, then apply this algebraic theory to the class of 3-cycle graphs known as friendship graphs. Specifically, we will use the algebra 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
We analyze the design of a mechanism to extract a ranking of individuals according to a unidimensional characteristic, such as ability or need. Individuals, connected on a social network, only have local information about the ranking. We show that a planner can construct an ex post incentive compatible and efficient mechanism if and only if every pair of friends has a friend in common. We characterize the windmill network as the sparsest social network for which the planner can always construct a complete ranking. (JEL D11, D82, D83, D85, O12, Z13)
We give character free proofs of two solvability theorems due to Isaacs.
For , 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 with the property that for each
subset of size , the set is a hyperedge.
We show that for , the number of hyperedges in a friendship
r-hypergraph is at least , and we
characterise those hypergraphs which achieve this bound. This generalises a
result given by Li and van Rees in the case when .
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.
Suppose that G is a graph with n vertices and m edges, and let μ be the spectral radius of its adjacency matrix.Recently we showed that if G has no 4-cycle, then μ2-μ⩽n-1, with equality if and only if G is the friendship graph.Here we prove that if m⩾9 and G has no 4-cycle, then μ2⩽m, with equality if G is a star. For 4⩽m⩽8 this assertion fails.
The 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
From rather modest beginnings, the British Combinatorial Conference grew into an established biennial international gathering. A successful format for the series of conferences was established, whereby several distinguished mathematicians were invited to give a survey lecture and to write a paper for the conference volume. The 1983 conference was held in Southampton, and this volume contains the invited papers, comprising three each from the United Kingdom, continental Europe and the United States. These papers cover a broad range of combinatorial topics, including enumeration, finite geometries, graph theory and permanents. The book will be of value not only to mathematicians, but also to scientists, engineers and others interested in combinatorial ideas.