  In this paper, we study dual quaternion and dual complex unit gain graphs and
their spectral properties in a unified frame of dual unit gain graphs. Unit
dual quaternions represent rigid movements in the 3D space, and have wide
applications in robotics and computer graphics. Dual complex numbers found
application in brain science recently. We establish the interlacing theorem for
dual unit gain graphs, and show that the spectral radius of a dual unit gain
graph is always not greater than the spectral radius of the underlying graph,
and these two radii are equal if and only if the dual gain graph is balanced.
By using the dual cosine functions, we establish the closed form of eigenvalues
of adjacency and Laplacian matrices of dual complex and quaternion unit gain
cycles. We then show the coefficient theorem holds for dual unit gain graphs.
Similar results hold for the spectral radius of the Laplacian matrix of the
dual unit gain graph too.
