A Random Walk Proof of Matrix Tree Theorem

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Outline• Kirchhoff’s Matrix Tree Theorem• Random Walk on a Graph• Wilson’s Algorithm• Proof of Wilson’s Algorithm and Kirchhoff’s Matrix Tree Theorem• Application: Cayley’s Formula1
Set-upSuppose that Γ = (V,E) is a finite graph consisting of n+1 vertices labelledy 1 ,y 2 ,··· ,y n ,y n+1 .• undirected• connected• no multiple edgesNote that y i ∼ y j are nearest neighbours if (y i ,y j ) ∈ E.y 1 y 5y 4y 3 y 2y 62
Page 1: A Random Walk Proof of Matrix Tree
Page 5 and 6: Kirchhoff’s Matrix Tree TheoremSu
Page 7 and 8: Example (cont.)This graph has 29 sp
Page 9 and 10: Random Walk on a GraphFormally, a s
Page 11 and 12: The Key Random Walk FactsLet ζ ∆
Page 13 and 14: Example: Wilson’s Algorithm on Γ
Page 15 and 16: Example: Wilson’s Algorithm on Γ
Page 17 and 18: Suppose ∆ ⊂ V, ∆ ≠ ∅.Comp
Page 19 and 20: Computing a Loop-Erased Walk Probab
Page 21 and 22: Proof of Wilson’s AlgorithmRecall
Page 23 and 24: Proof of Wilson’s AlgorithmRecall
Page 25 and 26: Application: Cayley’s FormulaIf
Page 27 and 28: Application: Cayley’s FormulaSinc
Page 29: Thank you.28

A Random Walk Proof of Matrix Tree Theorem

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