The Structure of Cayley Graph of Generalized Quaternion Group of Valency Less than or Equal 3

Let G be a group and S be a subset of G such that S excludes the identity element and is closed under taking inverses. The graph Cay(G, S) is a simple Cayley graph in which the vertices correspond to the elements of G and two vertices are adjacent if and only if one can be obtained by multiplying the other by an element from S. Cay(G, S) is a |S|-regular. This study is motivated by the work of Al-Kaseasbeh and Erfanian (2021), who explored the structure of Cayley graphs of dihedral groups with valencies 1, 2, and 3. Like dihedral groups, generalized quaternion groups are generated by two elements. In this research, the author aims to analyze the structure of Cayley graphs of the generalized quaternion group with valency at most 3.