A proof of Vince's conjecture on the average order of connected induced subgraphs

We prove that the average order of connected induced subgraphs of a connected simple graph of order $n$ and minimum degree at least $3$ is strictly greater than $n\mathrm{/}2$, which settles a conjecture of Vince~[\emph{J. Graph Theory} 97 (2021), 82–103]. We go further and show that, for connected simple graphs of order $n$ and minimum degree exactly $k\ge 3$, $K_k\vee (K_1\cup K_{n-k-1})$ uniquely minimizes this average under additional conditions.