The Zero-divisor Graph of      ℤ n  [ X ]]

Min Ji Park; Eun Sup Kim; Jung Wook Lim

doi:10.5666/KMJ.2020.60.4.723

Article

Kyungpook Mathematical Journal 2020; 60(4): 723-729

Published online December 31, 2020

The Zero-divisor Graph of ℤ n [ X ]]

Min Ji Park, Eun Sup Kim, Jung Wook Lim^*

Department of Mathematics, College of Life Science and Nano Technology, Hannam University, Daejeon 34430, Republic of Korea
e-mail : mjpark5764@gmail.com
Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 41566, Republic of Korea
e-mail : eskim@knu.ac.kr
Department of Mathematics, College of Natural Sciences, Kyungpook National University, Daegu 41566, Republic of Korea
e-mail : jwlim@knu.ac.kr

Received: June 13, 2020; Revised: July 28, 2020; Accepted: August 4, 2020

Abstract

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Abstract

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

Keywords: Γ(ℤ,_n[X)⟧,, diameter, girth, clique, chromatic number