Amenable Graphs: Theory and Applications

• Amenable graphs are defined as locally finite graphs with a vanishing vertex-isoperimetric constant, allowing large finite subsets with negligible boundaries.
• They exhibit close ties with automorphism group properties and unimodularity, linking classical and weighted amenability to spectral and percolative analyses.
• Amenable graphs impact diverse areas—from phase transitions in statistical physics to harmonic functions in planar maps—highlighting their broad mathematical significance.

An amenable graph is, in the classical graph-theoretic sense used throughout much of the modern literature, a locally finite graph whose vertex-isoperimetric constant vanishes: if $\partial A$ denotes the vertex boundary of a finite set $A$, then $$h(\Gamma)=\inf_{\substack{A\subseteq V(\Gamma)\ |A|<\infty}} \frac{|\partial A|}{|A|}, \qquad \Gamma \text{ is amenable if } h(\Gamma)=0.$$ In a closely related formulation used for infinite quasi-transitive graphs, one writes $$\partial A:=\{x\in A:\exists y\notin A,\ \{x,y\}\in E\}$$ and again requires the corresponding boundary-to-volume infimum to be $0$ (Tessera et al., 2022, Raoufi, 2016). This is a Følner-type condition: one can find large finite sets whose boundary is negligible compared with their volume. In transitive and quasi-transitive settings, amenability is tightly linked to automorphism groups, while in nonunimodular settings it is replaced by weighted-amenability defined from the Haar modulus (Terlov et al., 4 Feb 2025). The term also has several specialized uses—in graph isomorphism, path algebras, and automorphism-group dynamics—so the subject is best treated as a family of notions organized around small-boundary geometry, hyperfiniteness, and spectral or probabilistic rigidity (Arvind et al., 2015).

1. Classical amenability and automorphism groups

For connected locally finite quasitransitive graphs, amenability is expressed through vanishing boundary growth and is equivalent to the existence of finite sets $A_n\subseteq V(\Gamma)$ with $\frac{|\partial A_n|}{|A_n|}\to 0$ (Tessera et al., 2022). The same source also introduces $r$-amenability, defined by the condition that for every $\varepsilon>0$ there exists a finite vertex set $F$ with $A$0, where $A$1. For bounded-degree graphs, ordinary amenability implies $A$2-amenability for every $A$3 (Tessera et al., 2022).

A central structural theorem identifies amenability of a quasitransitive graph with amenability and unimodularity of the acting automorphism group. If $A$4 is connected and locally finite and $A$5 is a locally compact group acting properly quasitransitively on $A$6, then $A$7 In particular, for the full automorphism group, a connected locally finite quasitransitive graph is amenable if and only if its automorphism group is amenable and unimodular (Tessera et al., 2022). The same paper reformulates this through geometric amenability, a right Følner condition on the acting locally compact group.

A related action-theoretic perspective replaces graphs by actions preserving a locally finite metric. In that setting, faithful amenable actions preserving a locally finite metric are exactly those arising from embeddings into amenable totally disconnected locally compact groups, and the transitive subclass is characterized by the existence of a co-amenable almost normal subgroup with trivial core (Anantharaman-Delaroche, 2018). In the graph case, this places amenability of a vertex action and amenability of the closure inside $A$8 into the same framework.

2. Weighted amenability and nonunimodular extensions

For a connected locally finite graph $A$9 together with a closed transitive subgroup $$h(\Gamma)=\inf_{\substack{A\subseteq V(\Gamma)\ |A|<\infty}} \frac{|\partial A|}{|A|}, \qquad \Gamma \text{ is amenable if } h(\Gamma)=0.$$0, nonunimodularity produces a canonical weight on vertices through the Haar modulus. If $$h(\Gamma)=\inf_{\substack{A\subseteq V(\Gamma)\ |A|<\infty}} \frac{|\partial A|}{|A|}, \qquad \Gamma \text{ is amenable if } h(\Gamma)=0.$$1 are in the same orbit, the relative weight is $$h(\Gamma)=\inf_{\substack{A\subseteq V(\Gamma)\ |A|<\infty}} \frac{|\partial A|}{|A|}, \qquad \Gamma \text{ is amenable if } h(\Gamma)=0.$$2 where $$h(\Gamma)=\inf_{\substack{A\subseteq V(\Gamma)\ |A|<\infty}} \frac{|\partial A|}{|A|}, \qquad \Gamma \text{ is amenable if } h(\Gamma)=0.$$3 is a left Haar measure on $$h(\Gamma)=\inf_{\substack{A\subseteq V(\Gamma)\ |A|<\infty}} \frac{|\partial A|}{|A|}, \qquad \Gamma \text{ is amenable if } h(\Gamma)=0.$$4. Fixing a root $$h(\Gamma)=\inf_{\substack{A\subseteq V(\Gamma)\ |A|<\infty}} \frac{|\partial A|}{|A|}, \qquad \Gamma \text{ is amenable if } h(\Gamma)=0.$$5, one writes $$h(\Gamma)=\inf_{\substack{A\subseteq V(\Gamma)\ |A|<\infty}} \frac{|\partial A|}{|A|}, \qquad \Gamma \text{ is amenable if } h(\Gamma)=0.$$6 (Terlov et al., 4 Feb 2025).

The corresponding notion of weighted-amenability is the weighted Følner condition $$h(\Gamma)=\inf_{\substack{A\subseteq V(\Gamma)\ |A|<\infty}} \frac{|\partial A|}{|A|}, \qquad \Gamma \text{ is amenable if } h(\Gamma)=0.$$7 The same paper introduces levels, defined by equality of weight, and slices, defined as subgraphs induced by finite unions of levels. The main equivalence theorem states that, for a transitive closed subgroup $$h(\Gamma)=\inf_{\substack{A\subseteq V(\Gamma)\ |A|<\infty}} \frac{|\partial A|}{|A|}, \qquad \Gamma \text{ is amenable if } h(\Gamma)=0.$$8, the following are equivalent: $$h(\Gamma)=\inf_{\substack{A\subseteq V(\Gamma)\ |A|<\infty}} \frac{|\partial A|}{|A|}, \qquad \Gamma \text{ is amenable if } h(\Gamma)=0.$$9 is $$\partial A:=\{x\in A:\exists y\notin A,\ \{x,y\}\in E\}$$0-amenable, $$\partial A:=\{x\in A:\exists y\notin A,\ \{x,y\}\in E\}$$1 is level-amenable, and $$\partial A:=\{x\in A:\exists y\notin A,\ \{x,y\}\in E\}$$2 is hyperfinite (Terlov et al., 4 Feb 2025). A foundational theorem recalled there is that $$\partial A:=\{x\in A:\exists y\notin A,\ \{x,y\}\in E\}$$3 is amenable if and only if $$\partial A:=\{x\in A:\exists y\notin A,\ \{x,y\}\in E\}$$4 is $$\partial A:=\{x\in A:\exists y\notin A,\ \{x,y\}\in E\}$$5-amenable, so weighted-amenability is the graph-theoretic avatar of amenability of the acting automorphism group.

This weighted theory has both spectral and percolative characterizations. The paper defines a canonical $$\partial A:=\{x\in A:\exists y\notin A,\ \{x,y\}\in E\}$$6-biased random walk and proves a weighted Kesten theorem: if $$\partial A:=\{x\in A:\exists y\notin A,\ \{x,y\}\in E\}$$7 denotes its spectral radius, then $$\partial A:=\{x\in A:\exists y\notin A,\ \{x,y\}\in E\}$$8 It also proves that $$\partial A:=\{x\in A:\exists y\notin A,\ \{x,y\}\in E\}$$9-amenability implies $0$0, while $0$1-nonamenability yields a Pak–Smirnova-Nagnibeda style relaxation under which a quasi-isometric transitive graph on the same vertex set satisfies $0$2. A further consequence is continuity at the heavy-cluster threshold: if $0$3 is $0$4-nonamenable, then under Bernoulli bond percolation at $0$5 there is no heavy cluster almost surely (Terlov et al., 4 Feb 2025).

A second weighted theory arises from dynamical extensions of Markov maps. For a weighted graph $0$6, the paper defines $0$7 by retaining only edges of weight $0$8 and calls $0$9 $A_n\subseteq V(\Gamma)$0-amenable if $A_n\subseteq V(\Gamma)$1 For topologically transitive extensions of Gibbs-Markov maps with full branches and uniform loops, this weighted amenability is equivalent to the statement that the spectral radius of the associated transfer operator $A_n\subseteq V(\Gamma)$2 is equal to $A_n\subseteq V(\Gamma)$3; in the symmetric case it is also equivalent to non-exponential decay of return probabilities (Jaerisch et al., 2024).

3. Planar graphs, harmonic functions, and large-scale rigidity

For planar maps, amenability is intertwined with potential theory. A locally finite planar map is non-amenable if there exists $A_n\subseteq V(\Gamma)$4 such that $A_n\subseteq V(\Gamma)$5 for every finite vertex set $A_n\subseteq V(\Gamma)$6, equivalently if its Cheeger constant is positive (Carmesin et al., 2015). In this setting, the paper proves the rigidity statement $A_n\subseteq V(\Gamma)$7 for locally finite planar maps, or contrapositively: every locally finite non-amenable planar map admits a non-constant Dirichlet harmonic function (Carmesin et al., 2015). The key new notion is roundabout-transience, defined by replacing each vertex with a cycle of length $A_n\subseteq V(\Gamma)$8. The paper proves that every locally finite non-amenable planar map is roundabout-transient, and every locally finite roundabout-transient planar map admits a non-constant Dirichlet harmonic function.

This separates planar amenability from the general case. The paper explicitly notes that outside the planar setting, “Liouville $A_n\subseteq V(\Gamma)$9 amenable” is false; it cites Benjamini’s example of a bounded-degree non-amenable Liouville graph (Carmesin et al., 2015). A plausible implication is that, in the planar category, amenability is unusually rigid because boundary expansion, transience, and harmonic function theory can be coupled by duality and square-tiling methods.

Amenability also constrains coarse embeddings into hyperbolic geometry. For finitely generated amenable groups, regular or coarse embeddability into a hyperbolic group is equivalent to polynomial growth and hence, by Gromov’s theorem as recalled in the paper, to virtual nilpotence. More precisely, a finitely generated amenable group regularly or coarsely embeds into a hyperbolic group if and only if it is virtually nilpotent; the same paper shows that if an amenable compactly generated group regularly maps to $\frac{|\partial A_n|}{|A_n|}\to 0$0, with $\frac{|\partial A_n|}{|A_n|}\to 0$1 of polynomial growth degree $\frac{|\partial A_n|}{|A_n|}\to 0$2, then the source has polynomial growth degree at most $\frac{|\partial A_n|}{|A_n|}\to 0$3 (Tessera, 2020). This suggests that amenability is compatible with hyperbolic targets only in the polynomial-growth regime.

4. Phase transitions, spectra, self-avoiding walks, and inference

Amenability has direct consequences in statistical mechanics on infinite graphs. For the ferromagnetic Ising model on amenable quasi-transitive graphs with exponential growth, Raoufi proves continuity of spontaneous magnetization at the critical inverse temperature: $\frac{|\partial A_n|}{|A_n|}\to 0$4 The result is obtained by adapting Hutchcroft’s percolation argument to free-boundary Ising correlations and then invoking the Aizenman–Duminil-Copin–Sidoravicius criterion on amenable graphs; the paper’s abstract summarizes the outcome by saying that the model undergoes a second order phase transition (Raoufi, 2016).

In spectral theory, amenability enters through Følner exhaustions. A one-by-one exhaustion is a combinatorial condition that excludes finitely supported eigenfunctions of Laplace and Schrödinger operators. On graphs with a cocompact automorphism group and a Følner sequence, the paper proves a localization principle: absence of finitely supported eigenfunctions is equivalent to absence of all $\frac{|\partial A_n|}{|A_n|}\to 0$5-eigenfunctions and to continuity of the integrated density of states at the corresponding spectral value (Grigorchuk et al., 2021). In particular, any finitely generated indicable amenable group has a Cayley graph without eigenvalues, while the same finitely generated group may also admit another generating set whose adjacency operator has pure point spectrum (Grigorchuk et al., 2021).

For self-avoiding walks, the decisive extra structure is not amenability alone but the existence of a graph height function. The paper proves that every infinite finitely generated elementary amenable group has Cayley graphs with harmonic strong graph height functions, whereas the Cayley graph of the Grigorchuk group, which is amenable but not elementary amenable, has no graph height function (Grimmett et al., 2015). This is used to control locality of the connective constant, equality between connective constant and bridge growth, and existence of a terminating approximation algorithm for $\frac{|\partial A_n|}{|A_n|}\to 0$6 in the classes where height functions exist.

Amenability also changes the information–computation landscape in graph-based inference. For graph sequences converging locally weakly to an infinite random rooted graph that is almost surely anchored-amenable and tame, and after revealing an arbitrarily small fraction of vertex labels through a binary erasure channel, the paper proves that a local polynomial-time algorithm achieves risk asymptotically equal to the Bayes-optimal risk. The key mechanism is a point-to-boundary information bound whose right-hand side is proportional to $\frac{|\partial A_n|}{|A_n|}\to 0$7, so Følner geometry makes outside influence negligible (Alaoui et al., 2019). The same paper contrasts this with random regular graphs, where a gap persists.

5. Unimodular, Borel, and point-process constructions

Amenability also governs equivariant constructions on random graphs and graphings. For ergodic amenable unimodular random graphs that are one-ended almost surely, there exists a factor of iid spanning tree that has one end almost surely. In the quasi-transitive unimodular case this yields the characterization: a quasi-transitive unimodular graph is amenable and has $\frac{|\partial A_n|}{|A_n|}\to 0$8 end if and only if it has an invariant spanning tree with $\frac{|\partial A_n|}{|A_n|}\to 0$9 end (Timar, 2018). The result sharpens earlier “one or two ends” spanning-tree theorems by removing the residual two-ended alternative.

In descriptive combinatorics, amenability can make fractional matchings Borel. If $r$0 is a locally finite Borel graph that is componentwise quasi-transitive and amenable, then, in the intended form stated in the abstract, existence of a fractional perfect matching implies existence of a Borel fractional perfect matching. In particular, if a countable quasi-transitive amenable graph admits a fractional perfect matching, then its Bernoulli graph admits a Borel fractional perfect matching (Murray, 13 Jan 2025). The proof averages fractional perfect matchings over $r$1, using Salvatori’s theorem that quasi-transitive amenability implies amenability of the automorphism group.

For invariant point processes on locally compact groups, the relevant object is the Palm groupoid. The paper proves that factor thinnings, factor markings, and factor graphs correspond to Borel subsets or Borel functions on the Palm groupoid, and that for a unimodular group the Palm groupoid is probability-measure-preserving (Mellick, 2021). Its main amenability theorem states that if $r$2 is a locally compact, second countable, unimodular, noncompact group and $r$3 is a free and ergodic invariant point process on $r$4 of finite intensity, then $r$5 admits Cayley factor graphs of amenable discrete groups if and only if $r$6 is amenable; in that case it admits Cayley factor graphs of all finitely generated infinite amenable groups (Mellick, 2021). Thus amenability of the ambient group becomes exactly the condition under which the full class of finitely generated infinite amenable Cayley graphs can be realized as factor graphs.

6. Specialized and overloaded uses of the term

The expression “amenable graph” is not uniform across the literature. In graph isomorphism, a finite graph $r$7 is called amenable if color refinement distinguishes $r$8 from every non-isomorphic graph $r$9. This is a purely algorithmic notion: the paper proves that amenable graphs in this sense are recognizable in time $\varepsilon>0$0 and that every such graph is compact in Tinhofer’s sense, meaning that its fractional automorphisms polytope is integral (Arvind et al., 2015). This usage is terminologically standard in that subfield but unrelated to Følner amenability.

A second specialized usage appears in graph-associated rings. For a finite-vertex directed graph $\varepsilon>0$1, the path algebra $\varepsilon>0$2 is amenable if and only if either the smallest hereditary subset containing all vertices on cycles is not all of $\varepsilon>0$3, or $\varepsilon>0$4 contains an exclusive maximal cycle; exhaustive amenability is characterized by a stronger four-way alternative involving infinite emitters, accessible cycles, or an exclusive maximal cycle (Lorensen et al., 2024). Here amenability belongs to the algebra $\varepsilon>0$5, not to the graph as a metric or isoperimetric object.

A third usage concerns amenability of automorphism groups rather than amenability of graphs themselves. The paper on countable homogeneous directed graphs determines exactly which automorphism groups are amenable in the topological-dynamical sense and which are uniquely ergodic (Pawliuk et al., 2017). Likewise, the paper on weak amenability of graph products studies weak amenability of groups built from a finite simplicial graph; it explicitly notes that the defining graph is a commutation diagram and that the result is not about amenability of the graph as a graph (Reckwerdt, 2015).

Taken together, these variants show that “amenable graph” is an overloaded expression. In the classical geometric sense it refers to vanishing boundary expansion; in the nonunimodular transitive setting it becomes weighted-amenability; in specialized subfields it can refer to complete success of color refinement or to amenability of graph-associated algebras. The shared theme is not a single invariant but a recurring small-boundary or finite-approximation principle, realized combinatorially, measurably, spectrally, or algebraically depending on context.