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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2503.03421 |
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Table of Contents:
- 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.