Proceedings of the American Mathematical Society
Let 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.
Northshield, S. (1992). Cogrowth of Regular Graphs. Proceedings of the American Mathematical Society, 116 (1), 203-205.