The closeness eigenvalues of graphs

For a graphG with u, v is an element of V(G), denote by d(G)(u, v) the distance between u and v inG, which is the length of a shortest path connecting them if there is at least one path from u to v in G and is infinity otherwise. The closenessmatrix of a graph G is the |V (G)| x|V( G)| symmetric matrix (c(G)(u, v))u, v is an element of V (G, where c(G)(u, v) = 2(-d)G(u, v) if u not equal v and 0 otherwise. The closeness eigenvalues of a graph G are the eigenvalues of C(G). We determine the graphs for which the second largest closeness eigenvalues belong to (-infinity, a), where a approximate to -0.1571 is the second largest root of x(3) - (x)2 - 11/8x - 3/16 = 0. We also identify the n-vertex graphs with a closeness eigenvalue of multiplicity n- 1, n - 2 and n - 3, respectively.