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 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 -1-22,-1)∪(-1,0)∪(0,-1+22αmin, (where αmin≥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.