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The Friendship Theorem Dr. John S. Caughman Portland State University. Public Service Announcement. “Freshman’s Dream”. ( a+b ) p = a p +b p …mod p …when a, b are integers …and p is prime. Freshman’s Dream Generalizes!. (a 1 +a 2 +…+a n ) p =a 1 p +a 2 p +…+ a n p …mod p

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## The Friendship Theorem Dr. John S. Caughman Portland State University

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**The Friendship TheoremDr. John S. CaughmanPortland State**University**Public Service Announcement**“Freshman’s Dream” (a+b)p=ap+bp …mod p …when a, b are integers …and p is prime.**Freshman’s Dream Generalizes!**(a1+a2+…+an)p=a1p +a2p +…+anp …mod p …when a, b are integers …and p is prime.**a1*** * * 0 a2 * * A = 0 0 a3 * 0 0 0 a4 Freshman’s Dream Generalizes (a1+a2+…+an)p = a1p +a2p +…+anp tr(A) p = tr(Ap) (mod p)****** * * * * * * A = * * * * * * * * Freshman’s Dream Generalizes! tr(A) p = tr(Ap) (mod p) tr(A p)= tr((L+U)p) =tr(Lp +Up) = tr(Lp)+tr(Up)=0+tr(U)p = tr(A)p Note:tr(UL)=tr(LU) so cross terms combine , and coefficients =0 mod p.**The Theorem**If every pair of people at a party has precisely one common friend, then there must be a person who is everybody's friend.**Cheap Example**Nancy John Mark**Cheap Example of a Graph**Nancy John Mark**What a Graph IS:**Nancy John Mark**What a Graph IS:**Nancy Vertices! John Mark**What a Graph IS:**Nancy Edges! John Mark**What a Graph IS NOT:**Nancy John Mark**What a Graph IS NOT:**Loops! Nancy John Mark**What a Graph IS NOT:**Loops! Nancy John Mark**What a Graph IS NOT:**Nancy Directed edges! Mark John**What a Graph IS NOT:**Nancy Directed edges! Mark John**What a Graph IS NOT:**Nancy Multi-edges! John Mark**What a Graph IS NOT:**Nancy Multi-edges! John Mark**‘Simple’ Graphs…**Nancy • Finite • Undirected • No Loops • No Multiple Edges John Mark**The Theorem, Restated**Let G be a simple graph with n vertices. If every pair of vertices in G has precisely one common neighbor, then G has a vertex with n-1 neighbors.**The Theorem, Restated**Generally attributed to Erdős (1966). Easily proved using linear algebra. Combinatorial proofs more elusive.**Pigeonhole Principle**If more than n pigeons are placed into n or fewer holes, then at least one hole will contain more than one pigeon.**Some threshold results**If a graph with n vertices has > n2/4 edges, then there must be a set of 3 mutual neighbors. If it has > n(n-2)/2 edges, then there must be a vertex with n-1 neighbors.**Extremal Graph Theory**If this were an extremal problem, we would expect graphs with MORE edges than ours to also satisfy the same conclusion…**1**2**1**2 3**1**4 2 3**Of the 15 pairs, 3 have four neighbors in common and 12 have**two in common. So ALL pairs have at least one in common. But NO vertex has five neighbors!**Summary**If every pair of vertices in a graph has at least one neighbor in common, it mightnot be possible to remove edges and produce a subgraph in which every pair has exactly one common neighbor.**Accolades for Friendship**• The Friendship Theorem is listed among Abad's “100 Greatest Theorems” • The proof is immortalized in Aigner and Ziegler's Proofs from THE BOOK.**How to prove it:**STEP ONE: If x and y are not neighbors, they have the same # of neighbors. Why: Let Nx = set of neighbors of x Let Ny = set of neighbors of y**y**x How to prove it:**y**x Nx How to prove it:**y**x Ny How to prove it:

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