Farey-subgraphs and Continued Fractions

Abstract

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.