Structure of Cayley Graph Over Generalized Quaternion Group

Abstract

We investigate the structure of undirected Cayley graphs on generalized quaternion groups constructed via inverse-closed connection sets excluding the identity element. Through an analysis of small valencies from one to four, we generalize the structural properties to arbitrary valency. We establish a complete classification theorem, proving that all resulting Cayley graphs are isomorphic to circulant graphs, regular bipartite graphs, or the edge-disjoint union of these two structures. These findings provide a definitive characterization of Cayley graph structures on generalized quaternion groups and establish their connectivity and algebraic properties.