Shahid Rajaee Teacher Training University Journal of Discrete Mathematics and Its Applications 2981-0809 11 2 2026 06 01 Cayley graphs and G-graphs of gyro-groups 107 122 12565 10.22061/jdma.2025.12144.1138 EN Gholam Hossein Fath-Tabar Department of Mathematics, Statistics and Computer Science, Faculty of Science, University of Kashan, Kashan 87317-51167, I. R. Iran; Neda Moradi Department of Mathematics, Statistics and Computer Science, Faculty of Science, University of Kashan, Kashan 87317-51167, I. Alain Bretto Universite de Caen, GREYC CNRS UMR-6072, Campus II Bd Marechal Juin BP 5186, 14032 Caen cedex Caen, France Journal Article 2025 06 07 The present paper investigates the structural properties of Cayley graphs and <em>G</em>-graphs associated with certain gyro-groups, providing rigorous proofs for several key characteristics. Additionally, we conduct a comprehensive review of specific classes of gyro-groups, including gyro-commutative gyro-groups, dihedral gyro-groups, and dihedralized gyro-groups. Subsequently, we derive and establish significant properties of the corresponding <em>G</em>-graphs. The study culminates in an examination of the symmetry properties exhibited by the Cayley graphs and <em>G</em>-graphs of selected gyro-groups, contributing to a deeper understanding of their algebraic and combinatorial structures. https://jdma.sru.ac.ir/article_12565_cb75dd11e67d4a7a422498a33c057131.pdf