1 / 15

Cayley’s Formula

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

rance
Download Presentation

Cayley’s Formula

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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)’ • |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 n=1 and n=2 • For some k in {1,2,3,…,n} there exists a dk such that dk=1reason: 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

  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 • 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 vi= (no. of appearances of ‘i’ in S)+1degree 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 edgesT 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)

  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.orgMathworld: www.mathworld.wolfram.com • http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Cayley.html

More Related