Browsing Mathematics Faculty Work by Subject "covering tree"
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Cogrowth of Regular GraphsLet G be a d-regular graph and T the covering tree of G. We define a cogrowth constant of G in T and express it in terms of the first eigenvalue of the Laplacian on G. As a corollary, we show that the cogrowth constant is as large as possible if and only if the first eigenvalue of the Laplacian on G is zero. Grigorchuk's criterion for amenability of finitely generated groups follows.
