Eccentricity spectral properties of $\mathcal{C}$-graphs

A cograph is a simple graph that contains no induced path on four vertices. In this paper, we consider $\mathcal{C}$-graphs, which are a specific class of cographs, defined as $$\overline{\overline{\overline{K_{\alpha_{1}}}\cup K_{\alpha_{2}}}\cup \cdots \cup K_{\alpha_{2k}}},$$ %\text{ where } k \geq 2, \alpha_{2k}\geq 2,$$ where $k \geq 2$, $\alpha_{2k} \geq 2$, and $K_{\alpha_{i}}$ denotes the complete graph on $\alpha_{i}$ vertices. We investigate the spectral properties of the eccentricity matrix of this particular class of cographs. Additionally, we determine the irreducibility and inertia of the eccentricity matrix of $\mathcal{C}$-graphs. Furthermore, we identify an interval $(-1-\sqrt{2},-2)\cup (-2,0)$ in which these graphs have no eccentricity eigenvalues.