A Generalized Information-Theoretic Approach for Bounding the Number of Independent Sets in Bipartite Graphs

• Main Author: Igal Sason
• Format: Article
• Language: English
• Published: MDPI AG 2021-02-01
• Series: Entropy
• Subjects: Shannon entropy; Shearer’s lemma; counting; independent sets; graphs.
• Online Access: https://www.mdpi.com/1099-4300/23/3/270

This paper studies the problem of upper bounding the number of independent sets in a graph, expressed in terms of its degree distribution. For bipartite regular graphs, Kahn (2001) established a tight upper bound using an information-theoretic approach, and he also conjectured an upper bound for general graphs. His conjectured bound was recently proved by Sah et al. (2019), using different techniques not involving information theory. The main contribution of this work is the extension of Kahn’s information-theoretic proof technique to handle irregular bipartite graphs. In particular, when the bipartite graph is regular on one side, but may be irregular on the other, the extended entropy-based proof technique yields the same bound as was conjectured by Kahn (2001) and proved by Sah et al. (2019).