  In this paper, we establish the relation between classic invariants of graphs
and their integer Laplacian eigenvalues, focusing on a subclass of chordal
graphs, the strictly chordal graphs, and pointing out how their computation can
be efficiently implemented. Firstly we review results concerning general graphs
showing that the number of universal vertices and the degree of false and true
twins provide integer Laplacian eigenvalues and their multiplicities.
Afterwards, we prove that many integer Laplacian eigenvalues of a strictly
chordal graph are directly related to particular simplicial vertex sets and to
the minimal vertex separators of the graph.
