  Assumed to be undirected, simple, and connected are all of the graphs in this
study, and adjacency matrix $A$ serves as the associated matrix. In this paper
we show that it is possible to relate a creation sequence for a type of
cographs (we call it $\mathcal{C}$-graphs). Those cographs can be defined by a
finite sequence of natural numbers. Using that sequence we obtain the inertia
of the cograph under consideration. An extended eigenvalue-free set from
$(-1,0)$ to $\big{[}\frac{-1-\sqrt{2}}{2}, -1)\cup (-1, 0) \cup (0,
\frac{-1+\sqrt{2}}{2}\alpha_{min}\big{]}$, (where $\alpha_{min}\geq1$ is the
smallest integer of the creation sequence) is obtained for the cographs under
consideration. Additionally, an exact formula is found for the characteristic
polynomial.
