Finite-Horizon Value Spectra of Degree-Based Indices on Restricted Polyomino Chains

Degree-based topological indices are widely used in chemical graph theory as molecular structure descriptors. Understanding their attainable values and degeneracies over structured graph families can therefore provide information beyond conventional extremal bounds. In this paper, we determine the complete finite-horizon value spectrum of degree-based topological indices on restricted polyomino chains, which constitute lattice-based molecular graph models. Each chain is encoded by a link word and represented as a walk on a two-state weighted directed graph. When the alternating cycle has a strictly larger mean weight than the two self-loops, the index admits an exact decomposition into a linear bulk term, an endpoint contribution, and a positive penalty associated with defects in the link word. This representation characterizes all attainable values for a fixed chain length through feasible endpoint-defect data. We derive a closed formula for the multiplicity of each endpoint-defect class and, for exactly representable local weights, provide an exact finite algorithm that generates every distinct value, its deficit from the maximum, its total link word multiplicity, and all attaining classes, without enumerating the exponentially large set of link words. The number of distinct levels grows at most quadratically with the chain length. Under strict alternating dominance, its asymptotic growth is quadratic when the ratio of the two defect costs is irrational and linear when this ratio is rational. For the augmented Zagreb index, we compute the defect parameters explicitly, recover the known maximum, and determine the second-largest distinct value for every admissible chain length. We also obtain the parity-dependent extremal gap and characterize and enumerate all second-extremal link words.