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

Abstract

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 $$\overline{\overline{\overline{K_{\hat{I}{\pm }_1}}\cup K_{\hat{I}{\pm }_2}}\cup \cdots \cup K_{\hat{I}{\pm }_{2k}}},$$ %\text{ where } k \geq 2, α_{2k}\geq 2,$$ where $k\ge 2$, $\hat{I}{\pm }_{2k}\ge 2$, and $K_{\hat{I}{\pm }_i}$ denotes the complete graph on $\hat{I}{\pm }_i$ 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.