A note on self-adjoint extensions of the Laplacian on weighted graphs

Abstract

We study the uniqueness of self-adjoint and Markovian extensions of the Laplacian on weighted graphs. We first show that, for locally finite graphs and a certain family of metrics, completeness of the graph implies uniqueness of these extensions. Moreover, in the case when the graph is not metrically complete and the Cauchy boundary has finite capacity, we characterize the uniqueness of the Markovian extensions.

A remark on the uniqueness of Silverstein extensions of symmetric Dirichlet forms

“…Then it follows from [, Example 4.12] (see also [, Section 5.3, Example 1]) that the birth and death process generated by is not conservative if and only if . See , , for the conservativeness criterion in terms of the volume growth rate of random walks on more general weighted graphs. Remark (Locally finite weighted graphs) In addition to the condition on the function , we assume that if and only if . Then the triple is regarded as a (locally finite) weighted graph in the sense of , . We define the weighted degree of a vertex by We further define and …”

“…If we denote by the L 2 ‐generator of the Dirichlet form , then by , Proposition 3.3]. Moreover, Assumption (A) implies that is essentially self‐adjoint and has a unique Markovian extension ([, Theorems 2 and A.1]). Hence Theorem follows because the Markov uniqueness implies the Silverstein uniqueness by definition.…”

“…See [9], [12], [16] for the conservativeness criterion in terms of the volume growth rate of random walks on more general weighted graphs. Then the triple (G, w, m) is regarded as a (locally finite) weighted graph in the sense of [17], [19]. We define the weighted degree of a vertex x ∈ V by σ (x k−1 , x k ) x 0 , x 1 , x 2 , .…”

“…. , x n ∈ V with x 0 = x, x n = y, x k ∼ x k+1 , 0 ≤ k ≤ n − 1 (see, e.g., [9], [12], [15], [16], [17]). Then ρ is an intrinsic metric for (E, F).…”

A new discrete Hopf–Rinow theorem

“…As for a Hopf-Rinow theorem, first discrete versions have been proven in [Mil11] and [HKMW13]. The argument given in [Mil11] is based on length spaces in the sense of Burago-Burago-Ivanov [BBI01] and, while not mentioned explicitly, the length spaces in question are metric graphs associated to discrete graphs.…”