  Let $G$ be a simple graph and $v$ be a vertex of $G$. The triangle-degree of
$v$ in $G$ is the number of triangles that contain $v$. While every graph has
at least two vertices with the same degree, there are graphs in which every
vertex has a distinct triangle-degree. In this paper, we construct an infinite
family of graphs with this property. We also study the vertex degrees and size
of graphs with this property.
