### Citations

162 |
Difference equations, isoperimetric inequality and transience of certain random walks
- Dodziuk
- 1984
(Show Context)
Citation Context ... g if and only if there exists C > 0 such that 1c < f(x)/g(x) < C for all x. The following proposition is a standard fact about self-adjoint operators and appears, in a slightly less general form, in =-=[3]-=-. 8 Sam Northshield Proposition 2. Let λ ≥ 0. Then there exists h > 0 such that ∆uh ≥ λh if and only if inf f 〈f,∆uf〉/〈f, f〉 ≥ λ . Proof. Suppose that ∆u0h ≥ λh for some h > 0. Define ∇f([x, y]) = α(x... |

153 |
Random walks and percolation on trees
- Lyons
- 1990
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Citation Context ...ity d − 1 no longer represents the growth of T ; we define the growth number of T to be, in general, gr(T ) = lim sup n→∞ |Sn(o)| 1n . We note that our definition of gr(T ) differs from that in Lyons =-=[7]-=- (he uses the lim inf), but, under the additional hypothesis that G covers a finite graph, limn→∞ |Sn(o)| 1n exists and thus both definitions agree. A natural conjecture is that cogr(G) = gr(T ) if an... |

92 |
The Ihara-Selberg zeta function of a tree lattice
- Bass
(Show Context)
Citation Context ...here q(x) = u− u2 − 1−u2d(x) (which is constant when G is regular). The operator ∆u has long appeared (though not with this notation) in the literature on zeta functions for graphs. For example, Bass =-=[1]-=- was the first to prove: Z(u) ≡ ∏ C (1− u|C|)−1 = 1 (1− u2)rdet(∆u) , where Z, the zeta function of a finite graph, is the product over “prime” cycles C, and r is the Betti number of the graph. See al... |

91 |
Zeta functions of finite graphs and coverings
- Terras, Stark
(Show Context)
Citation Context ...irst to prove: Z(u) ≡ ∏ C (1− u|C|)−1 = 1 (1− u2)rdet(∆u) , where Z, the zeta function of a finite graph, is the product over “prime” cycles C, and r is the Betti number of the graph. See also papers =-=[9, 11, 6]-=- for other proofs of this generalization of Ihara’s theorem. Lemma 1 then states that ∆u is essentially the inverse of Ku. We say that a function is u-superharmonic if ∆uf ≥ 0. As in the usual case, H... |

48 | Spectra and function theory for combinatorial Laplacians - Dodziuk, Karp - 1988 |

39 | Zeta functions of finite graphs
- Kotani, Sunada
- 2000
(Show Context)
Citation Context ...irst to prove: Z(u) ≡ ∏ C (1− u|C|)−1 = 1 (1− u2)rdet(∆u) , where Z, the zeta function of a finite graph, is the product over “prime” cycles C, and r is the Betti number of the graph. See also papers =-=[9, 11, 6]-=- for other proofs of this generalization of Ihara’s theorem. Lemma 1 then states that ∆u is essentially the inverse of Ku. We say that a function is u-superharmonic if ∆uf ≥ 0. As in the usual case, H... |

23 | Symmetrical random walks on discrete groups - Grigorchuk - 1980 |

12 | Cogrowth of regular graphs
- Northshield
- 1992
(Show Context)
Citation Context ... and only if the number of words of length n in a coset grows as fast as the total number of words of length n in F grows. It was later noticed that this result can be extended to regular graphs (see =-=[8]-=-, for example). The concept of amenability was extended to graphs by Gerl: we say 2 Sam Northshield that a graph is amenable if and only if inf K |∂K| |K| = 0 , where the infimum is over all finite no... |

3 | Several Proofs of Ihara’s Theorem
- Northshield
- 1997
(Show Context)
Citation Context ...irst to prove: Z(u) ≡ ∏ C (1− u|C|)−1 = 1 (1− u2)rdet(∆u) , where Z, the zeta function of a finite graph, is the product over “prime” cycles C, and r is the Betti number of the graph. See also papers =-=[9, 11, 6]-=- for other proofs of this generalization of Ihara’s theorem. Lemma 1 then states that ∆u is essentially the inverse of Ku. We say that a function is u-superharmonic if ∆uf ≥ 0. As in the usual case, H... |

1 |
A note on recurrence, amenability, and the universal cover of graphs, in: Random Discrete Structures
- Northshield
- 1993
(Show Context)
Citation Context ...ghest possible Hausdorff dimension. Most of the terminology below appears in the seminal paper by Lyons [7]. The proof is based on the proof of the analogous fact for regular graphs which appeared in =-=[10]-=-. 12 Sam Northshield Proof of Theorem 3. We note that since G is a cover of a finite graph, T is also, and so it is “quasispherical”. A consequence is that limn→∞ |Sn| 1n exists. Fix k, and let R′ = {... |