On Limits of Vertex-Symmetric Graphs and Their Automorphisms
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Limits of Vertex-Symmetric Graphs and Their Automorphisms V. I. Trofimov1,2 Received September 19, 2019; revised October 15, 2019; accepted October 21, 2019
Abstract—Using a simple but rather general method of constructing Cayley graphs with trivial vertex stabilizers, we give an example of an infinite locally finite Cayley graph (and, hence, an example of an infinite connected locally finite vertex-symmetric unimodular graph) which is isolated in the space of connected locally finite vertex-symmetric graphs. We also give examples of Cayley graphs which are not isolated in this space but are isolated from the set of connected vertex-symmetric finite graphs. Keywords: connected locally finite vertex-symmetric graph, Cayley graph, convergence of graphs.
DOI: 10.1134/S0081543820040185 1. INTRODUCTION AND AUXILIARY RESULTS This note arose in connection with [1]. In it we show that, using simple observations concerning the convergence of connected locally finite vertex-symmetric graphs in combination with a method of construction of Cayley graphs with trivial vertex stabilizers as well as with some known results, one can find in the space of connected locally finite vertex-symmetric graphs an isolated infinite Cayley graph (and, consequently, an isolated infinite unimodular graph; this is an answer to the question from [1]) and a nonisolated Cayley graph, which is isolated from the set of connected vertex-symmetric finite graphs. In this note, a graph is an undirected graph without loops or multiple edges. If Γ is a graph, then V (Γ) is the set of its vertices, E(Γ) is the set of its edges, dΓ (·, ·) is the usual distance (a metric if Γ is connected) on V (Γ), and Aut(Γ) is the group of automorphisms of Γ (regarded as permutations on V (Γ)). Further, if x ∈ V (Γ), then Γ(x) = {y ∈ V (Γ) : {x, y} ∈ E(Γ)} is the neighborhood of x in Γ, BΓ (x, r) = {y ∈ V (Γ) : dΓ (x, y) ≤ r} for r ∈ R is the ball of radius r centered at x in the graph Γ, and Gx for G ≤ Aut(Γ) is the stabilizer of x in the group G. In addition, XΓ , where X ⊆ V (Γ), is the subgraph of Γ generated by X. A graph Γ is vertex-symmetric if Aut(Γ) is transitive on V (Γ). For a group G and its generating set M = M −1 1, denote by ΓG,M the Cayley graph of G with respect to M . Let G be the set of all connected locally finite vertex-symmetric graphs (considered up to Γ2 , the distance isomorphism) equipped with the following metric ρ(·, ·): for Γ1 , Γ2 ∈ G, Γ1 ∼ = 1
Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia 2 Ural Federal University, Yekaterinburg, 620002 Russia e-mail: trofi[email protected]
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between Γ1 and Γ2 is defined as ρ(Γ1 , Γ2 ) = 2−n , where n = max{m ∈ N ∪ {0} : BΓ1 (x1 , m)Γ1 ∼ = BΓ2 (x2 , m)Γ2 , x1 ∈ V (Γ1 ), x2 ∈ V (Γ2 )} (here and elsewhere, N is the set of positive integers). The metric space of connected locally finite vertex-symmetric graphs G, which appeared largely due to [2, 3], is essentially used in [4, 5]. In the rest of this section, we foll
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