A Random Walk Proof of Matrix Tree Theorem

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ExampleThis graph has 29 spanning trees.To see this, consider deg(y 4 ) in the spanning tree.y 1 y 5y 4y 3 y 2y 6deg(y 4 ) No. Spanning <strong>Tree</strong>s4 23 102 131 4295
Example (cont.)This graph has 29 spanning trees. For example, using MTT, det[L {5} ] = 29.y 1 y 5y 4y 3 y 2y 6L {5} =y 1 y 2 y 3 y 4 y 6⎡y 1 3 −1 −1 −1 0y 2 −1 2 −1 0 0y 3 −1 −1 4 −1 −1⎢y 4 ⎣ −1 0 −1 3 0y 6 0 0 −1 0 2⎤⎥⎦6
Page 1 and 2: A Random Walk Proof of Matrix Tree
Page 3 and 4: Set-upSuppose that Γ = (V,E) is a
Page 5: Kirchhoff’s Matrix Tree TheoremSu
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

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