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

2025-03-05

AI Summary

This paper investigates structural properties and coding applications of the unit graph $G(mathbb{Z}_n)$ over the ring $mathbb{Z}_n$, where $n$ is a specific positive integer. Using algebraic graph theory and ring-theoretic techniques, we prove that $G(mathbb{Z}_n)$ is connected with diameter at most 3 and determine its edge connectivity. Leveraging its incidence matrix, we construct a $q$-ary linear code and fully characterize its parameters—length, dimension, minimum distance, and dual code parameters. We resolve, for the first time, two central conjectures posed by Jain (2023) concerning the structure and coding-theoretic properties of unit graphs. This work establishes a systematic bridge among number theory, graph theory, and coding theory. Beyond advancing the theoretical understanding of unit graphs, it introduces a novel paradigm for graph-based coding over finite rings.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

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.

Problem

• Analyzes connectivity and diameter of unit graph G(ℤₙ).
• Constructs q-ary linear codes from G(ℤₙ) incidence matrix.
• Resolves conjectures on G(ℤₙ) structural and coding properties.

Innovation

• Constructs q-ary linear codes from unit graphs
• Analyzes connectivity and diameter of unit graphs
• Resolves conjectures on graph and coding properties