In this paper we consider particular graphs defined by $\overline{\overline{\overline{K_{α_1}}\cup K_{α_2}}\cup\cdots \cup K_{α_k}}$, where $k$ is even, $K_α$ is a complete graph on $α$ 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 $(α_1, α_2, \ldots, α_k)$. 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.