Abstract
In this paper, we extend our research on Cayley graphs of gyrogroups by exploring connections between the algebraic characteristics of gyrogroups and the combinatorial characteristics of their Cayley graphs. We look into some automorphisms of right Cayley graphs, give sufficient and necessary criteria for a right Cayley graph to be vertex-transitive by left gyrotranslations, and prove a similar result for quotient gyrogroups. Some sufficient and necessary criteria for a left gyroaddition to preserve edge color and for a gyration to be an automorphism on the graph are also given. Lastly, we show that certain Cayley graphs of a gyrogroup encode the gyroaddition table and provide an algorithm to compute the gyroaddition with the aid of the gyration table.
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