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The universal adjacency matrix $U$ of a graph $\Gamma$, with adjacency matrix$A$, is a linear combination of $A$, the diagonal matrix $D$ of vertex degrees,the identity matrix $I$, and the all-1 matrix $J$ with real coefficients, thatis, $U=c_1 A+c_2 D+c_3 I+ c_4 J$, with $c_i\in {\mathbb R}$ and $c_1\neq 0$.Thus, as particular cases, $U$ may be the adjacency matrix, the Laplacian, thesignless Laplacian, and the Seidel matrix. In this note, we show that basicallythe same method introduced before by the authors can be applied for determiningthe spectra and bases of all the corresponding eigenspaces of arbitrary liftsof graphs (regular or not).

A note on the spectra and eigenspaces of the universal adjacency matrices of arbitrary lifts of graphs