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 \(\left[ \frac{-1-\sqrt{2}}{2}, -1)\cup (-1, 0) \cup (0, \frac{-1+\sqrt{2}}{2}\alpha _{min}\right] \) , (where \(\alpha _{min}\ge 1\) 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.
