Counting independent sets in expanding bipartite regular graphs

TL;DR

This work studies i(G), the number of independent sets in d-regular bipartite graphs with vertex-expansion and bounded co-degree, via the independence polynomial Z(G, λ). It develops a defect polymer model and applies a cluster expansion to obtain a full asymptotic expansion of Z(G, λ) for λ in a broad range and derives sharp i(G) asymptotics, notably i(G) = 2^{n/2+1} exp(n/2^{d+1} + O(n d^2 / 2^{2d})). The results apply to well-known graph families such as the hypercube and middle-layer graphs and extend to Cartesian products of bounded-size base graphs, with new vertex-isoperimetry bounds for product graphs. The paper also provides an algorithmic framework to compute higher-order expansion terms and discusses the typical structure of weighted independent sets, contributing both to counting theory and practical approximation techniques in statistical physics-inspired models.

Abstract

In this paper we provide an asymptotic expansion for the number of independent sets in a general class of regular, bipartite graphs satisfying some vertex-expansion properties, extending results of Jenssen and Perkins on the hypercube and strengthening results of Jenssen, Perkins and Potukuchi. More precisely, we give an expansion of the independence polynomial of such graphs using a polymer model and the cluster expansion. In addition to the number of independent sets, our results yields information on the typical structure of (weighted) independent sets in such graphs. The class of graphs we consider covers well-studied cases like the hypercube or the middle layers graph, and we show further that it includes any Cartesian product of bipartite, regular base graphs of bounded size. To this end, we prove strong bounds on the vertex expansion of bipartite and regular Cartesian product graphs, which might be of independent interest.

Figures (5)

• Figure 1: Illustration of the proof of \ref{['prop:Icaptured']}. There is a (blue) vertex $v$ in $N(S) \cap [S']$ and a (red) vertex $u$ in $S$ whose neighbour is the vertex $v$. By definition of $[S']$ there exists a (dashed) edge between $S'$ and $S$.
• Figure 2: The defect configuration $I^{\mathcal{D}}= I \cap \mathcal{D}$ is chosen according to $\nu^{\mathcal{D}}$. In $\mathcal{D}'$ vertices in the neighbourhood of $I^{\mathcal{D}}$ are blocked. $I \cap \mathcal{D}'$ includes every unblocked vertex, i.e., vertices in $\mathcal{D}' \setminus N(I^{\mathcal{D}})$, independently with probability $\frac{\lambda}{1+\lambda}$.
• Figure 3: A cluster in $G$ with defect side $\mathcal{O}$. The cluster $\Gamma$ depicted here consists of two polymers of size $2$ and twice the same polymer of size one. The incompatibility graph $H(\Gamma)$ is depicted on the left side.
• Figure 4: From left to right the three possibilities for the structure of a cluster of size $2$ and the corresponding incompatibility graphs.
• Figure 5: Computation times for first coefficients

Theorems & Definitions (32)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3: Kotecký--Preiss condition
• Lemma 2.4
• Lemma 2.5: a version of Lemma 1.2 in JeMaPa24
• Proposition 3.1
• proof