Independent Sets in Graphs with Given Minimum Degree

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The enumeration of independent sets in graphs with various restrictions has been a topic of much interest of late. Let $i(G)$ be the number of independent sets in a graph $G$ and let $i_t(G)$ be the number of independent sets in $G$ of size $t$. Kahn used entropy to show that if $G$ is an $r$-regular bipartite graph with $n$ vertices, then $i(G)\leq i(K_{r,r})^{n/2r}$. Zhao used bipartite double covers to extend this bound to general $r$-regular graphs. Galvin proved that if $G$ is a graph with $\delta(G)\geq \delta$ and $n$ large enough, then $i(G)\leq i(K_{\delta,n-\delta})$. In this paper, we prove that if $G$ is a bipartite graph on $n$ vertices with $\delta(G)\geq\delta$ where $n\geq 2\delta$, then $i_t(G)\leq i_t(K_{\delta,n-\delta})$ when $t\geq 3$. We note that this result cannot be extended to $t=2$ (and is trivial for $t=0,1$). Also, we use Kahn's entropy argument and Zhao's extension to prove that if $G$ is a graph with $n$ vertices, $\delta(G)\geq\delta$, and $\Delta(G)\leq \Delta$, then $i(G)\leq i(K_{\delta,\Delta})^{n/2\delta}$.