Universal spectra of the disjoint union of regular graphs

A universal adjacency matrix of a graph $G$ with adjacency matrix $A$ is any matrix of the form $U=\alpha A+\beta I+\gamma J+\delta D$ with $\alpha \neq 0$,where$I$istheidentitymatrix,$J$istheall-onesmatrixand$D$isthediagonalmatrixwiththevertexdegrees\mathrm{.}Inthecasethat$G$ is the disjoint union of regular graphs, we present an expression for the characteristic polynomials of the various universal adjacency matrices in terms of the characteristic polynomials of the adjacency matrices of the components. As a consequence we obtain a formula for the characteristic polynomial of the Seidel matrix of $G$, and the signless Laplacian of the complement of $G$ (i.e. the join of regular graphs).