Farey-Subgraphs and Continued Fractions

Abstract

In this article, we study a family of subgraphs of the Farey graph, denoted as ℱN for every N ∈ ℕ. We show that ℱN 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 ℱN -continued fractions for each N > 1. We establish a relation between ℱN-continued fractions and certain paths from infinity in the graph ℱN. Using this correspondence, we discuss the existence and uniqueness of ℱN-continued fraction expansions of real numbers.