  A graph is called "Laplacian integral" if the eigenvalues of its Laplacian
matrix are all integers. We investigate the subset of these graphs whose
Laplacian is furthermore diagonalized by a matrix with entries coming from a
fixed set, with particular emphasis on the sets $\{-1,0,1\}$ or $\{-1,1\}$.
Such graphs include as special cases the recently-investigated families of
"Hadamard-diagonalizable" and "weakly Hadamard-diagonalizable" graphs. As a
combinatorial tool to aid in our investigation, we introduce a family of
vectors that we call "balanced", which generalize totally balanced partitions,
regular sequences, and complete partitions. We show that balanced vectors
completely characterize which graph complements and complete multipartite
graphs are $\{-1,0,1\}$-diagonalizable, and we furthermore prove results on
diagonalizability of the Cartesian product, disjoint union, and join of graphs.
Particular attention is paid to the $\{-1,0,1\}$- and
$\{-1,1\}$-diagonalizability of the complete graphs and complete multipartite
graphs. Finally, we provide a complete list of all simple, connected graphs on
nine or fewer vertices that are $\{-1,0,1\}$- or $\{-1,1\}$-diagonalizable.
