Weighted Harary indices of apex trees and k-apex trees

If G is a connected graph, then HA(G) = ∑ u6=v(deg(u) + deg(v))/d(u, v) is the additively Harary index and HM (G) = ∑ u6=v deg(u) deg(v)/d(u, v) the multiplicatively Harary index of G. G is an apex tree if it contains a vertex x such that G − x is a tree and is a k-apex tree if k is the smallest integer for which there exists a k-set X ⊆ V (G) such that G − X is a tree. Upper and lower bounds on HA and HM are determined for apex trees and k-apex trees. The corresponding extremal graphs are also characterized in all the cases except for the minimum k-apex trees, k ≥ 3. In particular, if k ≥ 2 and n ≥ 6, then HA(G) ≤ (k+ 1)(3n− 5n−k−k+ 2)/2 holds for any k-apex tree G, equality holding if and only if G is the join of Kk and K1,n−k−1.