On Pauling's residual entropy estimate for regular graphs with growing degree

TL;DR

The paper proves that for sequences of $d$-regular graphs with growing degree and under weak constraints on short closed trails, Pauling's residual-entropy estimate $\widehat{\rho}(G)$ is asymptotically exact, i.e., $\rho(G)=\widehat{\rho}(G)+o(1)$. The authors express $\rho(G)$ as $\widehat{\rho}(G)+\frac{1}{n}\log\mathbb{E}2^{|\mathcal{T}(\boldsymbol{P})|}$ over Eulerian partitions and bound the contribution of closed trails by splitting into long and short trails. Long trails are controlled via moment-generating bounds derived from a distribution $X(m)$, while short trails are bounded using a general switching theorem (Sissy's framework) applied to Eulerian partitions, with length-$\ell$ switchings and a cardinality bound $c_{k,\ell}(G) \le C e^{-(\ell+1)} d^{\ell-1} n$. The results yield corollaries for graphs with spectral constraints, growing girth, and Cartesian products such as hypercubes; collectively these establish the conjecture in broad families, including several classical constructions. The work provides a robust combinatorial-toolkit (switchings plus Hölder decompositions) for controlling entropy contributions from small substructures in large regular graphs.

Abstract

In 1935, Pauling proposed an estimate for the number of Eulerian orientations of a graph in the context of the theoretical behaviour of water ice. The logarithm of the number of Eulerian orientations, normalised by the number of vertices, is called the residual entropy. In an earlier paper, we conjectured that the residual entropy of a sequence of regular graphs of increasing degree was asymptotically equal to Pauling's estimate. Here we prove the conjecture under constraints on the number of short circuits. These constraints hold under weak eigenvalue conditions and apply to sequences of increasing girth and repeated Cartesian products such as hypercubes.

Figures (2)

• Figure 1: A graph $G$ and an Eulerian partition $P \in \mathcal{P}(G)$.
• Figure 2: Example of the action of a switching at one vertex. Simultaneous similar actions at all vertices visited by the trail $T$ transform Eulerian partition $P$ into $P'$. Pairs $(e_1, e_1')$ and $(e_2, e_2')$ are in $T$. The dotted edge pair in $P$ is not involved in the switching and remains in $P'$.

Theorems & Definitions (25)

• Conjecture 1.1: Special case of IMZ1 for regular graphs
• Theorem 2.1
• Corollary 2.2
• Corollary 2.3
• Corollary 2.4
• proof : Proof of Corollary \ref{['C:spectral']}
• proof : Proof of Corollary \ref{['C:spectral2']}
• proof : Proof of Corollary \ref{['C:product']}
• Lemma 3.1
• proof