On the third largest eigenvalue of eccentricity matrices of graphs

Let D(G) denote the distance matrix of a connected graph G. The eccentricity matrix (or anti-adjacency matrix) of G is obtained from D(G) 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 Kt ∨ (G ∪ kK1), where k ⩾ 0 and G is a chain graph with at most ten cells.