2025/03/05 by Sarkar, Apurba, Hansda, Kalyan, Maji, Makhan
#06F25 #13M99 #94B05 #Commutative Algebra (math.AC) #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT)
paper · doi:10.48550/arxiv.2503.03421
In this paper, we study the unit graph G(ℤn) , where n is of the form n = p1n1 p2n2 … prnr, with p1, p2, …, pr being distinct prime numbers and n1, n2, …, nr being positive integers. We establish the connectivity of G(ℤn) , 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(ℤn) , explicitly determining their parameters and duals. A primary contribution of this work is the resolution of two conjectures from \citeJain2023 concerning the structural and coding-theoretic properties of G(ℤn) . These results extend the study of algebraic graph structures and highlight the interplay between number theory, graph theory, and coding theory.