Extremal degree-based indices of general polyomino chains via dynamic programming

TL;DR

This paper introduces a dynamic programming method to identify extremal general polyomino chains with respect to degree-based indices, solving an open problem about maximizing the generalized Randić index for any number of squares.

Contribution

The paper presents a novel dynamic programming framework for extremal problems in polyomino chains, explicitly solving an open problem from 2015.

Findings

Maximized the generalized Randić index for polyomino chains with any number of squares.

Extremal configurations depend on the number of squares modulo 4.

The approach provides a systematic method for extremal graph theory problems.

Abstract

In this paper, we develop a dynamic programming framework for identifying extremal general polyomino chains with respect to degree-based topological indices. As a concrete application, we resolve an open problem posed in 2015 by determining, for any given number of squares, the general polyomino chains that maximize the generalized Randi\'c index with parameter $\alpha=-1$. We show that the extremal configurations depend explicitly on the residue class of the number of squares modulo 4. Beyond this specific result, the proposed dynamic programming approach provides a constructive and systematic methodology for tackling extremal problems in graph theory.