Degree-based topological indices of monogenic semigroup graphs: reduced Zagreb indices, coindices and PI index

Topological indices play a fundamental role in the study of graph structures and their quantitative properties, particularly in applications related to chemistry, network science, and dynamic system analysis. In this work, we study the Reduced Zagreb index, Zagreb coindices, and the PI index of monogenic semigroup graphs, a well-studied class of algebraic graphs with relevance to both pure and applied mathematics. Using the adjacency structure of the graph as the main tool, we derive exact closed-form expressions for these indices and illustrate the results through explicit examples. Furthermore, we obtain explicit parity-dependent closed-form formulas for the reduced first Zagreb index, reduced second Zagreb index, Zagreb coindices and the vertex-PI index of monogenic semigroup graphs. The derivations are based on a neighbourhood-block decomposition of the edge set, which provides a unified computational framework for several families of degree-based topological indices. These results demonstrate how the algebraic structure of a monogenic semigroup determines the behaviour of important graph-theoretical descriptors.