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Cayley’s Formula
- Srinivas Nambirajan
The Setting
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Arthur Cayley (August 16, 1821 – January 26, 1895)
Pure Mathematician
Group Theory (Cayley’s Theorem)
Matrices (Cayley-Hamilton Theorem)
Trinity College, Cambridge
The Formula
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Statement:
‘The number of distinct trees possible, on a set of n labelled vertices is n(n-2)’
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|Tn| = n(n-2)
The Methods
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Induction
Direct
The Intense
The Outline
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Claim:
For a set of n labelled vertices {V} and a set of n positive integers {d} such
that
, let d(vi) = di. Then
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Proof by Induction
From |A| to |T|
Multinomial Theorem to arrive at final expression
The Inductive Step
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Claim true for n=1 and n=2
For some k in {1,2,3,…,n} there exists a dk such that dk=1
reason: degree sum<2n (formal proof, using A.M.>=G.M)
Since {d} is a fixed degree sequence, k, once chosen is fixed
k=n, say.
Inductive hypothesis: |V|=n-1.
|Bi| is number of distinct trees on {v1, v2, …, vn-1}
db= degree of vb = di if b != i
= di-1 if b=i
|A| is the sum over all possible |Bi|
Proof of claim follows
The ‘Multinomial’ Step
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|A| is for a specific degree sequence summing up to 2(n-1)
|T| is the sum of all |A| over..
Multinomial theorem:
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Proof follows
The Elegant
The Outline
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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
The Sequence
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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
The Bijection
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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 vi= (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, vn) or (vn,vn-1)
The Equivalent
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Number of S such that number of entries in S is n-2
n ways to fill up each entry
Proof follows
The Prufer Way
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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
The End (Bibliography)
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Wikipedia: www.wikipedia.org
Mathworld: www.mathworld.wolfram.com
http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html