1. Cayley’s Formula - Srinivas Nambirajan

2. The Setting Arthur Cayley (August 16, 1821 – January 26, 1895) Pure Mathematician Group Theory (Cayley’s Theorem) Matrices (Cayley-Hamilton Theorem) Trinity College, Cambridge

3. The Formula Statement: ‘The number of distinct trees possible, on a set of n labelled vertices is n (n-2) ’ |T n | = n (n-2)

4. The Methods Induction Direct

5. The Intense

6. The Outline Claim: For a set of n labelled vertices {V} and a set of n positive integers {d} such that, let d(v i ) = d i. Then Proof by Induction From |A| to |T| Multinomial Theorem to arrive at final expression

7. The Inductive Step Claim true for n=1 and n=2 For some k in {1,2,3,…,n} there exists a d k such that d k =1 reason: degree sum =G.M) Since {d} is a fixed degree sequence, k, once chosen is fixed k=n, say. Inductive hypothesis: |V|=n-1. |B i | is number of distinct trees on {v 1, v 2, …, v n-1 } d b = degree of v b = d i if b != i = d i -1 if b=i |A| is the sum over all possible |B i | Proof of claim follows

8. The ‘Multinomial’ Step |A| is for a specific degree sequence summing up to 2(n-1) |T| is the sum of all |A| over.. Multinomial theorem: Proof follows

9. The Elegant

10. The Outline Represent a tree T in terms of a sequence of numbers S such that S  T Problem translates to finding number of such sequences given a vertex set

11. The Sequence For a tree, remove the lowest among the end vertices in any given step For every removal, write down the index of the node to which the removed vertex is attached to Proceed till 2 vertices are left Terminate sequence Example: Sequence: 4445

12. The Bijection S  T T=>S (If not, then the tree has no end vertices in some step => No vertices exist or a cycle exists) All ‘S’es lead to a tree: degree of a vertex v i = (no. of appearances of ‘i’ in S)+1 degree sum=no. of terms in sequence + 1 for every vertex in vertex set = n-2+n = 2n-2 = 2(n-1) = 2(e) e = n-1 Uniqueness: S to T: A sequence gives all the n-1 edges T to S: ambiguity => cycle (contrapositive) S is a representation of n-2 ordered pairs (comparison set) Ordered pair => edge n-2 edges known. Last edge given by end vertices. End vertices (last entry, v n ) or (v n,v n-1 )

13. The Equivalent Number of S such that number of entries in S is n-2 n ways to fill up each entry Proof follows

14. The Prufer Way S is a Prufer Sequence Heinz Prufer: German mathematician Nothing to do with ketchup Heinz is like ‘Bob’ in Germany Devised the idea to prove Cayley’s formula in 1918

15. The End (Bibliography) Wikipedia: www.wikipedia.org Mathworld: www.mathworld.wolfram.comwww.wikipedia.orgwww.mathworld.wolfram.com http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html