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 $$\stackrel{‾}{\stackrel{‾}{\overline{K_{{\alpha }_1}}\cup K_{{\alpha }_2}}\cup \cdots \cup K_{{\alpha }_{2k}}},$$ %$\text{ where }$ k \geq 2, α_{2k}\geq 2,$$where$k \geq 2$,$α_{2k} \geq 2$,and$K_{α_{i}}$denotesthecompletegraphon$α_{i}$vertices\mathrm{.}Weinvestigatethespectralpropertiesoftheeccentricitymatrixofthisparticularclassofcographs\mathrm{.}Additionally,wedeterminetheirreducibilityandinertiaoftheeccentricitymatrixof$$\mathcal{C}$$-graphs\mathrm{.}Furthermore,weidentifyaninterval$(-1-\sqrt{2},-2)\cup (-2,0)$ in which these graphs have no eccentricity eigenvalues.