Arithmetic Divisor Graph on \(\mathbb{Z}_n\)
R. S. Indu *
Department of Mathematics, University College, Thiruvananthapuram, Kerala-695034, India.
A. V. Vismaya
Department of Mathematics, University College, Thiruvananthapuram, Kerala-695034, India.
*Author to whom correspondence should be addressed.
Abstract
We introduce a new graph structure, called the arithmetic divisor graph GAD(\(\mathbb{Z}_n\)), defined on \(\mathbb{Z}_n\) 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(\(\mathbb{Z}_n\)): 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(\(\mathbb{Z}_n\)) 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 results
reveal a strong interaction between additive and multiplicative structures in \(\mathbb{Z}_n\), providing a new perspective on arithmetic graphs.
Keywords: Ring of integers, units, Hamiltonian graph