We introduce a new graph structure, called the arithmetic divisor graph GAD (\ (Zₙ\) ), defined on \ (Zₙ\) by declaring two vertices adjacent whenever the difference of their standard representatives divides their sum modulo n. We establish several fundamental structural properties of GAD (\ (Zₙ\) ): it is connected with radius 1, and it is complete if and only if n ≤ 4, while for n > 4 it has diameter 2. We further show that GAD (\ (Zₙ\) ) is Hamiltonian for all n ≥ 3, and we investigate its degree bounds, clique structure, and adjacency behavior. In particular, we obtain necessary conditions for adjacency among units, expressed in terms of congruence restrictions modulo prime divisors of n, leading to a complete characterization for odd moduli. Our resultsreveal a strong interaction between additive and multiplicative structures in \ (Zₙ\), providing a new perspective on arithmetic graphs.
Indu et al. (Mon,) studied this question.
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