  Let $\mathcal{E}$ be a $\mathbb{Q}$-isogeny class of elliptic curves defined
over $\mathbb{Q}$. The isogeny graph associated to $\mathcal{E}$ is a graph
which has a vertex for each element of $\mathcal{E}$ and an edge for each
$\mathbb{Q}$-isogeny of prime degree that maps one element of $\mathcal{E}$ to
another element of $\mathcal{E}$, with the degree recorded as a label of the
edge. The isogeny-torsion graph associated to $\mathcal{E}$ is the isogeny
graph associated to $\mathcal{E}$ where, in addition, we label each vertex with
the abstract group structure of the torsion subgroup over $\mathbb{Q}$ of the
corresponding elliptic curve. The main result of the article is a determination
of which isogeny-torsion graphs associated to $\mathbb{Q}$-isogeny classes of
elliptic curves defined over $\mathbb{Q}$ correspond to infinitely many
$\textit{j}$-invariants.
