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In this paper we give a simple characterization of the Laplacian spectra of a family of graphs as the eigenvalues of symmetric tridiagonal matrices. In addition, we apply our result to obtain upper and lower bounds for the Laplacian-energy-like invariant of these graphs. The class of graphs considered are obtained from copies of modified generalized Bethe trees (obtained by joining the vertices at some level by paths), identifying their roots with the vertices of a regular graph or a path.