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The ABS (atom-bond sum-connectivity) index of a graph G is denoted by A B S (G) and is defined as ∑ x y ∈ E (G) (d x + d y) − 1 (d x + d y − 2) , where d x represents the degree of the vertex x in G. In this paper, we derive the best possible upper bounds on the ABS index for fixed-order trees possessing a given maximum degree, which provides a solution to the open problem proposed quite recently by Hussain, Liu and Hua. [ABSTRACT FROM AUTHOR]

Cruz, Monsalve and Rada [Extremal values of vertex-degree-based topological indices of chemical trees, Appl. Math. Comput. 380 (2020) 125281] posed an open problem to find the maximum value of the exponential second Zagreb index for chemical trees of given order. In this paper, we solve the open problem completely. [ABSTRACT FROM AUTHOR]