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

Abstract

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.