Maximum augmented Zagreb index on polyomino chains

In this paper, we present a dynamic programming approach for identifying extremal polyomino chains with respect to degree–based indices. This approach provides an explicit recurrence, and a constructive algorithm that enable both the computation of an extremal polyomino chain in linear time with respect to the number of squares, and the enumeration of all maximal configurations. As a main application, we focus on the problem posed in 2016 by characterizing the polyomino chains that maximize the Augmented Zagreb Index ( AZI ) for any fixed number of squares. The results presented in this paper are aligned with previous contributions, and establish a constructive methodology for solving extremal problems in chemical graph theory. A link to the implementation code is provided in the last section.