The diameter and eccentricity eigenvalues of graphs.

The eccentricity matrix ℰ (G) = ( u v) of a graph G is constructed from the distance matrix by keeping each row and each column only the largest distances with u v = d (u , v) , if d (u , v) = min { (u) , (v) } , 0 , otherwise , where d (u , v) is the distance between two vertices u and v , and (u) = max { d (u , v) | v ∈ V (G) } is the eccentricity of the vertex u. The ℰ -eigenvalues of G are those of its eccentricity matrix. In this paper, employing the well-known Cauchy Interlacing Theorem we give the following lower bounds for the second, the third and the fourth largest ℰ -eigenvalues by means of the diameter d of G : ξ 2 (G) ≥ − 1 , if d ≤ 2 ; α d , if d ≥ 3 , ξ 3 (G) ≥ − d , and ξ 4 (G) ≥ − 1 − 5 2 d , where α = 0. 3 1 1 1 + is the second largest root of x 3 − x 2 − 3 x + 1 = 0. Moreover, we further discuss the graphs achieving the above lower bounds. [ABSTRACT FROM AUTHOR]