Construction of Linear Codes from the Unit Graph \(G(\mathbb{Z}_{n})\)

In this paper, we consider the unit graph \(G(\mathbb{Z}_{n})\), where \(n=p_{1}^{n_{1}} \text{ or } p_{1}^{n_{1}}p_{2}^{n_{2}} \text{ or } p_{1}^{n_{1}}p_{2}^{n_{2}}p_{3}^{n_{3}}\) and \(p_{1}, p_{2}, p_{3}\) are distinct primes. For any prime \(q\), we construct \(q\)-ary linear codes from the incidence matrix of the unit graph \(G(\mathbb{Z}_{n})\) with their parameters. We also prove that the dual of the constructed codes have minimum distance either 3 or 4. Lastly, we stated two conjectures on diameter of unit graph \(G(\mathbb{Z}_{n})\) and linear codes constructed from the incidence matrix of the unit graph \(G(\mathbb{Z}_{n})\) for any integer \(n\).