Closed Forms and Bounds for Zagreb Indices of Dihedral Orbit Graphs

Abstract

In this scientific paper, we review the Zagreb indices for the orbital graphs of the D_2n group by highlighting some of the geometric properties of the orbital structure created from a dihedral group. We explain that when n is an odd number, the resulting graph is a complete graph. This structural description allows us to derive explicit and precise mathematical formulas for both indices. Furthermore, we prove the existence of a mathematical relationship between the two topological indices, showing that the second Zagreb index is expressed mathematically in terms of the first Zagreb index through a multiplicative constant that depends on the order of the orbital graph. The obtained results summarize the strength of the relationship and interaction between the group structure and the indices, establishing a smooth framework for studying these indices in algebraically defined orbital graphs.