Weighted Harary indices of apex trees and kk-apex trees

If GG is a connected graph, then HA(G)=∑u≠v(deg(u)+deg(v))/d(u,v)HA(G)=∑u≠v(deg(u)+deg(v))/d(u,v) is the additively Harary index and HM(G)=∑u≠vdeg(u)deg(v)/d(u,v) the multiplicatively Harary index of GG. GG is an apex tree if it contains a vertex xx such that G−xG−x is a tree and is a kk-apex tree if kk is the smallest integer for which there exists a kk-set X⊆V(G)X⊆V(G) such that G−XG−X is a tree. Upper and lower bounds on HAHA and HMHM are determined for apex trees and kk-apex trees. The corresponding extremal graphs are also characterized in all the cases except for the minimum kk-apex trees, k≥3k≥3. In particular, if k≥2k≥2 and n≥6n≥6, then HA(G)≤(k+1)(3n2−5n−k2−k+2)/2HA(G)≤(k+1)(3n2−5n−k2−k+2)/2 holds for any kk-apex tree GG, equality holding if and only if GG is the join of KkKk and K1,n−k−1K1,n−k−1.