Abstract
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).
Highlights
The Shannon entropy and other classical information measures serve as a powerful tool in various combinatorial and graph-theoretic applications, such as the method of types, applications of Shearer’s lemma, sub- and supermodularity properties of information measures and their applications, entropy-based proofs of Moore bound for irregular graphs, Bregman’s theorem on the permanent of square matrices with binary entries, and a discrepancy theorem by Spencer
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 that was conjectured by Kahn [11] and proved by Sah et al [23]
The bound is achieved by a disjoint finite union of such complete bipartite graphs, since the number of independent sets in a disjoint union of graphs is equal to the product of the number of independent sets in each of these component graphs
Summary
The Shannon entropy and other classical information measures serve as a powerful tool in various combinatorial and graph-theoretic applications (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20]), such as the method of types, applications of Shearer’s lemma, sub- and supermodularity properties of information measures and their applications, entropy-based proofs of Moore bound for irregular graphs, Bregman’s theorem on the permanent of square matrices with binary entries, and a discrepancy theorem by Spencer. Upper bounding the number of independent sets in a regular graph was motivated in [21] by a conjecture which has several applications in combinatorial group theory. A decade later (2010), it was extended in [23] to regular graphs (that are not necessarily bipartite); a year later (2011), it was proved in [24] for graphs with a small maximal degree (up to 5) This conjecture was recently (2019) proved in general [25], by utilizing a new approach. 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 that was conjectured by Kahn [11] and proved by Sah et al [23].
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