The power graph P(G) of a group G is an undirected graph with
all the elements of G as vertices and where any two vertices u and v are
adjacent if and only if u=vm or v=um, m∈Z. For a
simple graph H with adjacency matrix A(H) and degree diagonal matrix
D(H), the universal adjacency matrix is U(H)=αA(H)+βD(H)+γI+ηJ, where α(î€ =0),β,γ,η∈R, I is
the identity matrix and J is the all-ones matrix of suitable order. One can
study many graph-associated matrices, such as adjacency, Laplacian, signless
Laplacian, Seidel etc. in a unified manner through the universal adjacency
matrix of a graph. Here we study universal adjacency eigenvalues and
eigenvectors of power graphs, proper power graphs and their complements on the
group Zn​, dihedral group Dn​, and the generalized quaternion
group Qn​. Spectral results of no kind for the complement of power graph on
any group were obtained before. We determine the full spectrum in some
particular cases. Moreover, several existing results can be obtained as very
specific cases of some results of the paper