Spectra of the blow-up graphs

Let $G$ be graph on $n$ vertices and $G^{(t)}$ its blow-up graph of order $t\mathrm{.}$ In this paper, we determine all eigenvalues of the Laplacian and the signless Laplacian matrix of $G^{(t)}$ and its complement $\overline{G^{(t)}}\mathrm{.}$