Generalized Pell graphs

In this paper, generalized Pell graphs ${\mathrm{\Pi }}_{n,k}$, $k\ge 2$, are introduced. The special case of $k=2$ are the Pell graphs ${\mathrm{\Pi }}_n$ defined earlier by Munarini. Several metric, enumerative, and structural properties of these graphs are established. The generating function of the number of edges of ${\mathrm{\Pi }}_{n,k}$ and the generating function of its cube polynomial are determined. The center of ${\mathrm{\Pi }}_{n,k}$ is explicitly described; if $k$ is even, then it induces the Fibonacci cube ${\mathrm{\Gamma }}_n$. It is also shown that ${\mathrm{\Pi }}_{n,k}$ is a median graph, and that ${\mathrm{\Pi }}_{n,k}$ embeds into a Fibonacci cube.