  In this paper we consider particular graphs defined by
$\overline{\overline{\overline{K_{\alpha_1}}\cup K_{\alpha_2}}\cup\cdots \cup
K_{\alpha_k}}$, where $k$ is even, $K_\alpha$ is a complete graph on $\alpha$
vertices, $\cup$ stands for the disjoint union and an overline denotes the
complementary graph. These graphs do not contain the $4$-vertex path as an
induced subgraph, i.e., they belong to the class of cographs. In addition, they
are iteratively constructed from the generating sequence $(\alpha_1, \alpha_2,
\ldots, \alpha_k)$. Our primary question is what invariants or graph properties
can be deduced form a given sequence. In this context, we compute the Lapacian
eigenvalues and the corresponding eigenspaces, and derive a lower and an upper
bound for the number of distinct Laplacian eigenvalues. We also determine the
graphs under consideration with a fixed number of vertices that either minimize
or maximize the algebraic connectivity (that is the second smallest Laplacian
eigenvalue). The clique number is computed in terms of a generating sequence
and a relationship between it and the algebraic connectivity is established.
