Asian Research Journal of Mathematics

This paper investigates the structural and algebraic properties of graphs generated by additive and multiplicative inverse operations modulo $n$. The study focuses on the interaction between the maps

\[
a \longmapsto-a \quad \text { and } \quad a \longmapsto a^{-1}
\]

on the group of units modulo\(n\). These transformations induce a possibly non-faithful action of the Klein four-group on the corresponding set of residue classes, providing an algebraic framework for analysing the structure of the associated graphs. Using group-action techniques and Burnside's lemma, we characterize the orbits arising from the combined action of negation and inversion and derive formulas for the number of resulting connected components. Particular attention is given to the fixed points of inversion and of the composition of negation and inversion, which are characterized by the congruences

\[
a^2 \equiv 1 \quad(\bmod n) \quad \text { and } \quad a^2 \equiv-1 \quad(\bmod n),
\]

respectively. These congruences play a central role in determining the orbit structure and edge configuration of the graphs. We further derive an explicit formula for the number of edges, including the necessary correction for edges arising simultaneously from the additive and multiplicative inverse operations.

The results provide a unified algebraic description of the structural properties of these modular graphs and demonstrate how the arithmetic properties of \(n\), particularly the solutions of quadratic congruences, govern their connectivity, orbit decomposition, and edge structure.