  In this paper, a new sub-family of Hypercubes called the \textit{associated
Mersenne graphs} $\mathcal{M}_{n}$ are introduced. The definition of associated
Mersenne graphs is motivated from the Fibonacci-run graphs ({\"O}.
E\v{g}ecio\v{g}lu, V. Ir\v{s}i\v{c}, 2021) by extending run-constrained strings
to circularly-run-constrained strings. The name of this new family of graphs is
identified with the interesting fact that $|V(\mathcal{M}_{n})|$ is equal to
the $n$-th associated Mersenne number. Various interesting structural and
enumerative properties of associated Mersenne graphs are investigated,
including the analogue of the fundamental recursion, number of vertices and
edges, radius, diameter, center, periphery and medianicity. Some future
research directions and open problems concerning associated Mersenne graphs are
also proposed.
