Counting independent sets in regular graphs with bounded independence number

An $n$-vertex, $d$-regular graph can have at most $2^{n/2+o_d(n)}$ independent sets. In this paper we address what happens with this upper bound when we impose the further condition that the graph has independence number at most $\alpha$. We give upper and lower bounds that in many cases are close to each other. In particular, for each $0 < c_{\rm ind} \leq 1/2$ we exhibit a constant $k(c_{\rm ind})$ such that if $(G_n)_{n \in {\mathbb N}}$ is a sequence of graphs with $G_n$ $d$-regular on $n$ vertices and with maximum independent set size at most $\alpha$, with $d\rightarrow \infty$ and $\alpha/n \rightarrow c_{\rm ind}$ as $n \rightarrow \infty$, then $G_n$ has at most $k(c_{\rm ind})^{n+o(n)}$ independent sets, and we show that there is a sequence $(G_n)_{n \in {\mathbb N}}$ of graphs with $G_n$ $d$-regular on $n$ vertices ($d \leq n/2$) and with maximum independent set size at most $\alpha$, with $\alpha/n \rightarrow c_{\rm ind}$ as $n \rightarrow \infty$ and with $G_n$ having at least $k(c_{\rm ind})^{n+o(n)}$ independent sets. We also consider the regime $1/2 < c_{\rm ind} < 1$. Here for each $0 < c_{\rm deg} \leq 1-c_{\rm ind}$ we exhibit a constant $k(c_{\rm ind},c_{\rm deg})$ for which an analogous pair of statements can be proven, except that in each case we add the condition $d/n \rightarrow c_{\rm deg}$ as $n \rightarrow \infty$. Our upper bounds are based on graph container arguments, while our lower bounds are constructive.