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 ? 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 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 ) ? ( k + 1 ) ( 3 n 2 - 5 n - k 2 - k + 2 ) / 2 holds for any k -apex tree G , equality holding if and only if G is the join of K k and K 1 , n - k - 1 .

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