Maximizing the Number of Independent Sets of a Fixed Size

Abstract

Letit(G) be the number of independent sets of sizetin a graphG. Engbers and Galvin asked how largeit(G) could be in graphs with minimum degree at least δ. They further conjectured that whenn⩾ 2δ andt⩾ 3,it(G) is maximized by the complete bipartite graphKδ,n−δ. This conjecture has recently drawn the attention of many researchers. In this short note, we prove this conjecture.