On Chemical Trees That Maximize Atom–Bond Connectivity Index, Its Exponential Version, and Minimize Exponential Geometric–Arithmetic Index (2020)
A chemical tree is a tree that has no vertex of degree greater than 4. We denote the set of chemical trees with n vertices as Cn. The ABC index of a chemical tree T is defined as ABC (T) = X 1≤i≤j≤4 mi,j (T) s i + j − 2 ij , where mi,j (T) is the number of edges in T joining vertices of degree i and j. Furtula, Graovac and Vukiˇcevi´c in 2009 found trees with maximal ABC index among all trees in Cn, when n ≡ 1 mod 4. In this paper we find the trees with maximal ABC index in Cn for all n. Using the same technique, we find the trees with maximal eABC and minimal e GA over Cn for all n, where eABC (T) = X 1≤i≤j≤4 mi,j (T) e qi+j−2 ij and e GA (T) = X 1≤i≤j≤4 mi,j (T) e 2 √ij i+j .