Lower Connective Constant in Graphs

• Lower connective constant is defined as the depth-wise infimum of exponential growth rates of self-avoiding walks in finite graphs, offering a refined measure of graph complexity.
• It aligns with the classical connective constant for infinite quasi-transitive graphs, enabling the transfer of infinite-volume analytical insights to finite-graph settings.
• Its applications include establishing rigorous lower bounds for models like the hard-core partition function and improving algorithmic approximations in statistical mechanics.

The lower connective constant is a self-avoiding-walk parameter introduced for families of finite graphs as a depth-wise infimum of exponential growth rates, and it is designed to capture graph complexity more precisely than maximum degree in analytic questions such as zero-freeness for the hard-core partition function. In the notation of the recent finite-graph formulation, it is defined from the numbers of self-avoiding walks of length at most $k$, while for infinite quasi-transitive graphs it is closely tied to the classical connective constant $\mu(G)$, the asymptotic growth rate of $n$-step self-avoiding walks. In related literature, the phrase also arises indirectly through universal lower bounds on $\mu(G)$ for regular, transitive, and vertex-transitive graphs (Chen et al., 3 Apr 2026, Grimmett et al., 2012, Grimmett et al., 2013).

1. Definitions and basic framework

For an infinite, connected, quasi-transitive graph $G$, the classical connective constant is

$\mu(G)=\lim_{n\to\infty} c_n(v)^{1/n},$

where $c_n(v)$ is the number of $n$-step self-avoiding walks starting at $v$. For vertex-transitive and quasi-transitive graphs, the limit exists and is independent of the starting vertex (Grimmett et al., 2013, Grimmett et al., 2014).

For finite graphs, the lower connective constant is defined differently. If $G$ is finite and $\mu(G)$0, let $\mu(G)$1 be the number of self-avoiding walks of length at most $\mu(G)$2 starting at $\mu(G)$3. Then

$\mu(G)$4

For a family $\mu(G)$5 of finite graphs,

$\mu(G)$6

This $\mu(G)$7 is the lower connective constant of the family (Chen et al., 3 Apr 2026).

Quantity	Formula	Setting
Connective constant	$\mu(G)$8	Infinite quasi-transitive graph
$\mu(G)$9-depth connective constant	$n$0	Finite graph
Lower connective constant	$n$1	Family of finite graphs

This finite-depth formulation is not merely terminological. It is constructed so that self-avoiding-walk growth can be used in finite-volume analytic arguments, especially when one wants uniform control over all members of a graph family (Chen et al., 3 Apr 2026).

2. Relation to the standard connective constant

A key property of the lower connective constant is its compatibility with the infinite-volume theory. For an infinite graph $n$2, if $n$3 denotes the family of finite induced subgraphs of $n$4, then the lower connective constant of $n$5 coincides with the standard connective constant $n$6 of the infinite graph (Chen et al., 3 Apr 2026).

This identification places the finite-depth notion squarely inside the classical theory of self-avoiding walks. In the older literature, the central object is always the exponential growth rate of self-avoiding walks, and many structural results are formulated directly for $n$7: existence via submultiplicativity, equality with bridge constants under suitable height-function hypotheses, and exact evaluation on a few special lattices (Grimmett et al., 2014, Grimmett et al., 2015, Duminil-Copin et al., 2010).

The finite-depth lower connective constant can therefore be viewed as a finite-graph proxy for $n$8. This suggests a transfer principle: results stated for graph families in terms of $n$9 recover infinite-lattice statements when applied to exhaustion families of induced subgraphs. That transfer is explicit in the zero-freeness theory for the hard-core model, where analyticity of the free energy on an infinite lattice is deduced from uniform zero-freeness over its finite induced subgraphs (Chen et al., 3 Apr 2026).

3. Universal lower bounds for the classical connective constant

Much of the literature surrounding “lower connective constant” concerns lower bounds for the standard connective constant on infinite regular graphs. Grimmett and Li proved that if $\mu(G)$0 and $\mu(G)$1 is an infinite, connected, $\mu(G)$2-regular, vertex-transitive graph, then

$\mu(G)$3

whenever $\mu(G)$4 is simple, or $\mu(G)$5 is non-simple and $\mu(G)$6 (Grimmett et al., 2012, Grimmett et al., 2013).

For cubic graphs, this gives

$\mu(G)$7

The bridge graph $\mu(G)$8 satisfies

$\mu(G)$9

so the bound is sharp in the class that allows multiple edges (Grimmett et al., 2012, Grimmett et al., 2013).

The proof strategy is combinatorial. It uses forward-extendable self-avoiding walks, meaning walks that can be extended to infinite self-avoiding walks, and shows that the number of forward-extendable walks of length $G$0 is at least $G$1. In the vertex-transitive simple case, a technical condition $G$2 is available and supports the counting argument. The same analysis is described in terms of “blue branches,” producing a recurrence that forces exponential growth at rate at least $G$3 (Grimmett et al., 2012, Grimmett et al., 2013).

These results delimit what can be asserted uniformly. For quasi-transitive graphs without vertex-transitivity, the best universal lower bound is only $G$4, and this is attained by regular graphs constructed by decorating an infinite line (Grimmett et al., 2012).

4. Exact values, extremal benchmarks, and the golden ratio problem

A small number of graphs admit exact connective constants, and these serve as benchmarks for lower-bound questions. Duminil-Copin and Smirnov proved that for the honeycomb lattice,

$G$5

using a parafermionic observable satisfying a half of the discrete Cauchy-Riemann relations (Duminil-Copin et al., 2010). Their method also established the corresponding critical point $G$6 through divergence and convergence of self-avoiding-walk generating functions (Duminil-Copin et al., 2010).

For the ladder graph $G$7,

$G$8

and this value is central in the theory of cubic graphs (Grimmett et al., 2012, Grimmett et al., 2017). Grimmett and Li explicitly raised the open question of whether every infinite, connected, cubic, vertex-transitive, simple graph satisfies

$G$9

At present, only the weaker rigorous bound $\mu(G)=\lim_{n\to\infty} c_n(v)^{1/n},$0 is known in full generality (Grimmett et al., 2013, Grimmett et al., 2017).

The Fisher transformation provides an extremal mechanism in the cubic setting. If $\mu(G)=\lim_{n\to\infty} c_n(v)^{1/n},$1 is cubic and $\mu(G)=\lim_{n\to\infty} c_n(v)^{1/n},$2 is obtained by replacing each degree-$\mu(G)=\lim_{n\to\infty} c_n(v)^{1/n},$3 vertex by a triangle, then for the iterated sequence $\mu(G)=\lim_{n\to\infty} c_n(v)^{1/n},$4, the connective constants satisfy

$\mu(G)=\lim_{n\to\infty} c_n(v)^{1/n},$5

and $\mu(G)=\lim_{n\to\infty} c_n(v)^{1/n},$6 decreases monotonically to $\mu(G)=\lim_{n\to\infty} c_n(v)^{1/n},$7 (Grimmett et al., 2017). A 2026 generalization replaces each degree-$\mu(G)=\lim_{n\to\infty} c_n(v)^{1/n},$8 vertex by a finite symmetric three-port gadget and obtains

$\mu(G)=\lim_{n\to\infty} c_n(v)^{1/n},$9

where $c_n(v)$0 is the two-port self-avoiding-walk generating function of the gadget (Grant et al., 18 Jan 2026). This shows that lower-bound problems are compatible with a broader local-transformation calculus than the Fisher triangle alone.

A weighted analogue appears in Glazman’s model on the dual $c_n(v)$1 lattice. For the critical family parametrized by $c_n(v)$2, the connective constant is

$c_n(v)$3

and the minimum over the family is attained at $c_n(v)$4, where it equals $c_n(v)$5 (Glazman, 2014).

5. Monotonicity, quotient operations, and locality

Lower bounds interact strongly with graph operations. Grimmett and Li proved strict inequalities for connective constants of vertex-transitive graphs: passing to a non-trivial quotient graph decreases the connective constant strictly, while adding a quasi-transitive family of edges increases it strictly (Grimmett et al., 2013). In the Cayley-graph setting, adding a new relator strictly decreases $c_n(v)$6, and declaring a non-trivial group element to be a generator strictly increases $c_n(v)$7 (Grimmett et al., 2013, Grimmett et al., 2013).

These monotonicity statements help organize extremal questions. If one seeks small connective constant within a structural class, quotienting and relator addition move in the lowering direction, whereas generator addition and edge augmentation move in the raising direction (Grimmett et al., 2013).

A complementary theme is locality. Grimmett and Li proved that if two infinite quasi-transitive graphs agree on a large ball around the origin, then their connective constants are close in value, provided the graphs admit unimodular graph height functions (Grimmett et al., 2014). The proof uses a generalized bridge decomposition, and under the same unimodularity hypothesis the bridge constant equals the connective constant: $c_n(v)$8 (Grimmett et al., 2014, Grimmett et al., 2015).

For Cayley graphs, the height-function framework becomes more algebraic. A group height function exists if and only if the coefficient matrix $c_n(v)$9 of the presentation satisfies $n$0 (Grimmett et al., 2015). Under an independent infinite-order generator hypothesis, local convergence of Cayley graphs implies convergence of connective constants: $n$1 (Song et al., 2014). This provides a rigorous approximation mechanism for connective constants by locally convergent graph sequences.

6. Algorithmic and analytic uses of the lower connective constant

The modern finite-graph notion of lower connective constant was introduced to improve degree-based analytic thresholds in the hard-core model. If a family $n$2 of finite graphs has lower connective constant $n$3, then for any $n$4, where

$n$5

the hard-core partition function is zero-free in a complex neighborhood of the interval $n$6 (Chen et al., 3 Apr 2026). The proof uses a block contraction technique that lifts correlation decay from a real interval to a strip-like complex neighborhood (Chen et al., 3 Apr 2026).

For an infinite lattice $n$7 with connective constant $n$8, the same framework yields uniqueness and analyticity of the free energy density up to the threshold $n$9 (Chen et al., 3 Apr 2026). Here the finite-family lower connective constant is the device that imports infinite-lattice connective-constant information into finite-volume zero-freeness.

Related algorithmic work had already shown that the ordinary connective constant governs spatial mixing and approximate counting. For graphs of bounded connective constant, Sinclair, Srivastava, Štefankovič, and Yin obtained strong spatial mixing and deterministic approximation schemes for the hard-core and monomer-dimer models, with thresholds parameterized by the connective constant rather than maximum degree (Sinclair et al., 2013, Sinclair et al., 2014). A later development replaces the global parameter by a local connective constant

$v$0

where $v$1 is the maximum number of $v$2-step self-avoiding walks from a vertex, and derives $v$3 mixing bounds for Glauber dynamics on the hard-core and Ising models (Efthymiou, 2024).

These developments show that self-avoiding-walk growth rates now function as algorithmic “effective degree” parameters. The lower connective constant is the version tailored to finite families and complex-analytic control, while the connective constant and local connective constant appear in correlation-decay and sampling results (Chen et al., 3 Apr 2026, Efthymiou, 2024).

7. Open problems and extensions

Two open problems remain prominent in the unweighted theory. The first is whether

$v$4

for every infinite, simple, transitive, cubic graph (Grimmett et al., 2013, Grimmett et al., 2017). The second is whether the lower bound

$v$5

extends to non-simple graphs when $v$6 (Grimmett et al., 2013).

Extensions to weighted self-avoiding walks broaden the scope of the subject. On finitely generated, virtually indicable groups, weighted connective and bridge constants coincide under a Hölder-type compatibility condition between the height function and the length function (Grimmett et al., 2018). By contrast, a 2025 matrix method for weighted self-avoiding walks develops systematic upper bounds via dominant eigenvalues and states explicitly that lower bounds are not the main focus, identifying them as a possible future direction (He, 4 Aug 2025).

A plausible implication is that “lower connective constant” now names a broader research program rather than a single invariant: finite-depth lower connective constants for graph families, universal lower bounds for $v$7, and weighted or local variants all address the same underlying issue, namely how much self-avoiding-walk entropy a graph must support under structural constraints.