Summary: D. Galvin [Discrete Math. 311, No. 20, 2105–2112 (2011; Zbl 1228.05228)] showed that for all fixed \(\delta \) and sufficiently large \(n\), the \(n\)-vertex graph with minimum degree \(\delta \) that admits the most independent sets is the complete bipartite graph \(K_{\delta, n-\delta}\). He conjectured that except perhaps for some small values of \(t\), the same graph yields the maximum count of independent sets of size \(t\) for each possible \(t\). Evidence for this conjecture was recently provided by J. Alexander et al. [Electron. J. Comb. 19, No. 3, Research Paper P37, 11 p. (2012; Zbl 1252.05157)] who showed that for all triples \((n,\delta, t)\) with \(t\geq 3\), no \(n\)-vertex bipartite graph with minimum degree \(\delta \) admits more independent sets of size \(t\) than \(K_{\delta, n-\delta}\). Here, we make further progress. We show that for all triples \((n,\delta, t)\) with \(\delta\leq 3\) and \(t\geq 3\), no \(n\)-vertex graph with minimum degree \(\delta \) admits more independent sets of size \(t\) than \(K_{\delta, n-\delta}\), and we obtain the same conclusion for \(\delta >3\) and \(t\geq 2\delta +1\). Our proofs lead us naturally to the study of an interesting family of critical graphs, namely those of minimum degree \(\delta \) whose minimum degree drops on deletion of an edge or a vertex.