The Zero-divisor Graph of the Ring of Integers Modulo n

Seung Jun Pi; Se Hun Kim; Jung Wook Lim∗

doi:10.5666/KMJ.2019.59.4.591

Article

KYUNGPOOK Math. J. 2019; 59(4): 591-601

Published online December 23, 2019

The Zero-divisor Graph of the Ring of Integers Modulo n

Seung Jun Pi, Se Hun Kim, Jung Wook Lim∗

Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu, 41566, Republic of Korea
e-mail : seungjunpi0@gmail.com
Department of Mathematics, Seoul National University, Seoul, 03080, Republic of Korea
e-mail : shunhun33@gmail.com
Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu, 41566, Republic of Korea
e-mail : jwlim@knu.ac.kr

Received: October 15, 2018; Revised: September 5, 2019; Accepted: November 11, 2019

Abstract

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Abstract

Let ℤ_n be the ring of integers modulo n and Γ(ℤ_n) the zero-divisor graph of ℤ_n. In this paper, we study some properties of Γ(ℤ_n). More precisely, we completely characterize the diameter and the girth of Γ(ℤ_n). We also calculate the chromatic number of Γ(ℤ_n).

Keywords: zero-divisor graph, ring of integers modulo n, diameter, girth, clique, chromatic number.