On the third largest eigenvalue of eccentricity matrices of graphs

Let $D\left(G\right)$ denote the distance matrix of a connected graph $G$. The eccentricity matrix (or anti-adjacency matrix) of $G$ is obtained from $D\left(G\right)$ by retaining in each row and each column only the maximal entries. In this paper, all the graphs with third largest eccentricity eigenvalue in the interval $(-1,0)$ are detected. It turns out that these graphs are all found among the chain graphs with (nonempty) four cells and the graphs of type $K_t\vee (G\cup k{K}_1)$, where $k⩾0$ and $G$ is a chain graph with at most ten cells.