A finite group's abstract structure is captured in a graph known as a Cayley graph. The name-bearing Cayley's theorem, which uses a preset set of generators for the group, is used to establish the definition. In 1878, Arthur Cayley began researching Cayley graphs for finite groups. In specific applications, such as the creation of interconnection networks for parallel CPUs, Cayley graphs are used. In this study, we analyse a subset S = {b, ab, (an-1)b} for dihedral groups of order 2n, where n>= 3, and determine the Cayley graph with respect to that subset. The respected Cayley graphs' eigenvalues and energies are also calculated.Cayley graphs are graphs connected to a group and a collection of generators for that group (there is also an associated directed graph). Due to their structure and symmetry, they provide great candidates for families of expander graphs. Additionally described is the unitary Cayley graph for detecting the Euler TOTIENT energy graph