On the spectra and eigenspaces of the universal adjacency matrices of arbitrary lifts of graphs

The universal adjacency matrix U of a graph Γ, with A, is linear combination the diagonal D vertex degrees, identity I, and all-1 J real coefficients, that is, U=c1A+c2D+c3I+c4J, ci∈R c1≠0. Thus, in particular cases, may be matrix, Laplacian, signless Seidel matrix. In this paper, we develop method for determining spectra bases all corresponding eigenspaces arbitrary lifts graphs (regular or not). As an example, applied to give efficient algorithm determine characteristic polynomial Laplacian symmetric squares odd cycles, together closed formulas some their eigenvalues.