On the number of connected sets in bounded degree graphs

A set of vertices in a graph is connected if it induces a connected subgraph. Using Shearer’s entropy lemma and a computer search, we show that the number of connected sets in a graph with n vertices and maximum vertex degree d is at most O(1.9351 ⁿ ) for d = 3, O(1.9812 ⁿ ) for d = 4, and O(1.9940 ⁿ ) for d = 5. Dually, we construct infinite families of graphs where the number of connected sets is at least Ω(1.7651 ⁿ ) for d = 3, Ω(1.8925 ⁿ ) for d = 4, and Ω(1.9375 ⁿ ) for d = 5.