Counting the number of dissociation sets in cubic graphs

Let $G$ be a graph. A dissociation set of $G$ is a subset of vertices that induces a subgraph with vertex degree at most 1. The dissociation polynomial of $G$ is $D_G(\lambda )=\sum _{D\in \mathcal{D}(G)}{\lambda }^{|D|}$, where $\mathcal{D}(G)$ is the set of all dissociation sets of $G$. In this paper, we prove that for any cubic graph $G$ and any $\lambda \in (0,1]$,

$$\frac{1}{|V(G)|}\ln D_G(\lambda )\le \frac{1}{4}\ln D_{K_4}(\lambda )$$

with equality if and only if $G$ is a disjoint union of copies of the complete graph $K_4$. When $\lambda =1$, the value of $D_G(\lambda )$ is exactly the number of dissociation sets of $G$. Hence, for any cubic graph $G$ on $n$ vertices, $|\mathcal{D}(G)|\le |\mathcal{D}(K_4){|}^{n/4}={11}^{n/4}.$