  In this note, we study a family of subgraphs of the Farey graph, denoted as
$\mathcal{F}_N$ for every $N\in\mathbb{N}.$
  We show that $\mathcal{F}_N$ is connected if and only if $N$ is either equal
to one or a prime power. We introduce a class of continued fractions referred
to as $\mathcal{F}_N$-continued fractions for each $N>1.$ We establish a
relation between $\mathcal{F}_N$-continued fractions and certain paths from
infinity in the graph $\mathcal{F}_N.$ We discuss existence and uniqueness of
$\mathcal{F}_N$-continued fraction expansions of real numbers.
