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 $n/2$, which settles a conjecture of Vince~[\emph{J. Graph Theory} 97 (2021), 82--103]. We go further and show that, for connected simple g