Weighted Harary indices of apex trees and k-apex trees

If G is a connected graph, then H_A(G)= _u≢v(deg(u)+deg(v))/d(u,v) is the additively Harary index and H_M(G)=_u≢vdeg(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 H_A and H_M 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 H_A(G) ≤ (k1)(3n² - 5n k² - k + 2) /2 holds for any k-apex tree G, equality holding if and only if G is the join of K_k and K₁,n-k-1).