An Entropy Approach to the Hard-Core Model on Bipartite Graphs
Combinatorics, Probability and Computing
On the Maximum Number of Cliques in a Graph
Graphs and Combinatorics
The number of independent sets in a regular graph
Combinatorics, Probability and Computing
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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 (2^d^+^1-1)^n^/^2^d by the Kahn-Zhao theorem [7,13]. Relaxing the regularity constraint to a minimum degree condition, Galvin [5] conjectured that, for n=2d, the number of independent sets in a graph with @d(G)=d is at most that in K"d","n"-"d. 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 Galvin@?s conjecture, covering the case n=