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

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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1. Public Service Announcement “Freshman’s Dream” (a+b)p=ap+bp …mod p …when a, b are integers …and p is prime.

2. Freshman’s Dream Generalizes! (a1+a2+…+an)p=a1p +a2p +…+anp …mod p …when a, b are integers …and p is prime.

3. 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)

4. * * * * * * * * 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.

5. 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.

6. Cheap Example Nancy John Mark

7. Cheap Example of a Graph Nancy John Mark

8. What a Graph IS: Nancy John Mark

9. What a Graph IS: Nancy Vertices! John Mark

10. What a Graph IS: Nancy Edges! John Mark

11. What a Graph IS NOT: Nancy John Mark

12. What a Graph IS NOT: Loops! Nancy John Mark

13. What a Graph IS NOT: Loops! Nancy John Mark

14. What a Graph IS NOT: Nancy Directed edges! Mark John

15. What a Graph IS NOT: Nancy Directed edges! Mark John

16. What a Graph IS NOT: Nancy Multi-edges! John Mark

17. What a Graph IS NOT: Nancy Multi-edges! John Mark

18. ‘Simple’ Graphs… Nancy • Finite • Undirected • No Loops • No Multiple Edges John Mark

19. 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.

20. The Theorem, Restated Generally attributed to Erdős (1966). Easily proved using linear algebra. Combinatorial proofs more elusive.

21. NOT A TYPICAL “THRESHOLD” RESULT

22. 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.

23. 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.

24. Extremal Graph Theory If this were an extremal problem, we would expect graphs with MORE edges than ours to also satisfy the same conclusion…

25. 1

26. 1 2

27. 1 2 3

28. 1 4 2 3

29. 4

30. 4

31. 4

32. 2

33. 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!

34. Related Fact – losing edges

35. Related Fact – losing edges

36. Related Fact – losing edges

37. 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.

38. 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.

39. Example 1

40. Example 2

41. Example 3

42. 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

43. y x How to prove it:

44. y x Nx How to prove it:

45. y x Ny How to prove it:

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