The number and average size of connected sets in graphs with degree constraints

The average size of connected vertex subsets a graph generalises much-studied parameter for subtrees trees. For trees, the possible values this are critically affected by presence or absence vertices degree 2. We answer two questions Andrew Vince regarding effect constraints on general graphs. give new lower bound, and first non-trivial upper maximum growth rate number sets cubic graph, in fact obtain bounds any constant bound degree. show that set density is bounded away from 1 graphs with no 2, generalise classical result Jamison trees showing order to approach 1, proportion 2 must 1. Finally, we sequence minimum tending infinity have 1/2.