A Random Walk Proof of Matrix Tree Theorem
A Random Walk Proof of Matrix Tree Theorem
A Random Walk Proof of Matrix Tree Theorem
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By the key random walk fact,Application: Cayley’s FormulaG ∆ (x,x) =11−r ∆ (x) = N(m+1)m(N +1) .(∗)Now, suppose that the vertices <strong>of</strong> Γ are {x 1 ,...,x N+1 }. Start the SRW at x 1 andassume that ∆ j = {x 1 ,...,x j } for j = 1,...,N.Since |∆ j | = j, we have from our linear algebra fact and (∗) thatdet[G {x 1} ] =N∏j=1G ∆j (x j ,x j ) =N∏j=1N(j +1)j(N +1) = NN (N +1)!(N +1) N N! = N N(N +1) N−1.Since each <strong>of</strong> the (N +1) vertices has degree N, we conclude|Ω| = det[D{x 1} ]det[G {x 1} ] =N N= (N +1) N−1 .N N(N+1) N−127
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