Linear Codes Derived from the Structure of Unit Graphs Over $\mathbb{Z}_n$

In this paper, we study the unit graph $G(\mathbb{Z}_n)$, where $n$ is of the form $n = p_1^{n_1} p_2^{n_2} \dots p_r^{n_r}$, with $p_1, p_2, \dots, p_r$ being distinct prime numbers and $n_1, n_2, \dots, n_r$ being positive integers. We establish the connectivity of $G(\mathbb{Z}_n)$, show that its diameter is at most three, and analyze its edge connectivity. Furthermore, we construct $q$-ary linear codes from the incidence matrix of $G(\mathbb{Z}_n)$, explicitly determining their parameters and duals. A primary contribution of this work is the resolution of two conjectures from \cite{Jain2023} concerning the structural and coding-theoretic properties of $G(\mathbb{Z}_n)$. These results extend the study of algebraic graph structures and highlight the interplay between number theory, graph theory, and coding theory.