Here, we analyze and compare properties of Cayley graphs of permutation graphs called transposition graphs as this family of graphs has better degree and diameter properties than the hypercube. First, the conjugacy class of the permutation group combined with its generators dictates the type of symmetry possessed by their respective Cayley graphs. In addition, the base-b hypercube of dimension n is a class of networks known to possess virtually every known notion of symmetry. Second, properties of the transposition graphs are given. Third, the spectrum of this class of graphs is discussed and formulas for the energy of these graphs are given as they are dependent on these properties. DOI: 10.1007/978-3-031-05375-7_23