It is known that the Cheeger constant of a hypercube graph $Q_n$ is exactly $1$, regardless of its dimension $n$. Is $1$ also an upper bound on the Cheeger constant of nontrivial induced connected subgraphs of $Q_n$?

The Cheeger constant $h(G)$ is also known as the edge expansion or the isoperimetic number.

To indirectly address a comment,
here are the Cheeger constants of more graphs:

$h($$Q_n$$) = 1$

$h($$P_n$$) = 1 / \lfloor n / 2 \rfloor$

$h($$C_n$$) = 2 / \lfloor n / 2 \rfloor$

$h($$K_n$$) = \lceil n / 2 \rceil$