The maximum number of complete subgraphs in a graph with given maximum degree.

Abstract: Extremal problems involving the enumeration of graph substructures have a long history in graph theory. For example, the number of independent sets in a d-regular graph on n vertices is at most by the Kahn–Zhao theorem [7,13]. Relaxing the regularity constraint to a minimum degree condition, Galvin [5] conjectured that, for , the number of independent sets in a graph with is at most that in . In this paper, we give a lower bound on the number of independent sets in a d-regular graph mirroring the upper bound in the Kahn–Zhao theorem. The main result of this paper is a proof of a strengthened form of Galvinʼs conjecture, covering the case as well. We find it convenient to address this problem from the perspective of . From this perspective, we show that the number of complete subgraphs of a graph G on n vertices with , where with , is bounded above by the number of complete subgraphs in . [Copyright &y& Elsevier]