Some lattice theoretical results on non-Euclidean graphs

This paper develops a unified framework connecting lattice theory and suborbital graphs, with particular focus on Farey graph. By equipping the Farey graph with lattice structures, we reveal new combinatorial and algebraic properties. Essential element graphs, Hasse diagrams, and integer sequences for vertices and edges are systematically explored. This study distinguishes itself from previous studies by expanding proofs, consolidating definitions, and providing illustrative examples. Our results demonstrate how a lattice perspective can enrich theoretical and applied research in number theory, geometry, and network science.