Cayleys Formula

1
Cayleys Formula

• - Srinivas Nambirajan

2
The Setting

• Arthur Cayley (August 16, 1821 January 26,
  1895)
• Pure Mathematician
• Group Theory (Cayleys Theorem)
• Matrices (Cayley-Hamilton Theorem)
• Trinity College, Cambridge

3
The Formula

• StatementThe number of distinct trees
  possible, on a set of n labelled vertices is
  n(n-2)
• Tn 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(vi) di. Then
• Proof by Induction
• From A to T
• Multinomial Theorem to arrive at final expression

7
The Inductive Step

• Claim true for n1 and n2
• For some k in 1,2,3,,n there exists a dk such
  that dk1reason degree sumlt2n (formal proof,
  using A.M.gtG.M)
• Since d is a fixed degree sequence, k, once
  chosen is fixed
• kn, say.
• Inductive hypothesis Vn-1.Bi is number of
  distinct trees on v1, v2, , vn-1 db degree
  of vb di if b ! i
  di-1 if bi
• A is the sum over all possible Bi
• 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 thatS?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
• ExampleSequence 4445

12
The Bijection

• S ? T
• TgtS (If not, then the tree has no end vertices
  in some step gt No vertices exist or a cylce
  exists)
• All Ses lead to a tree degree of a vertex vi
  (no. of appearances of i in S)1degree sumno.
  of terms in sequence 1 for every vertex in
  vertex set n-2n
  2n-2 2(n-1) 2(e)
• e n-1
• Uniqueness A sequence gives all the n-1 edges
• S is a representation of n-2 ordered pairs
  (comparison set)
• Ordered pair gt edge
• n-2 edges known. Last edge given by end vertices.
• End vertices (last entry, vn) or (vn,vn-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 Cayleys formula in 1918

15
The End (Bibliography)

• Wikipedia www.wikipedia.orgMathworld
  www.mathworld.wolfram.com
• http//www-groups.dcs.st-and.ac.uk/history/Mathem
  aticians/Cayley.html