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eISSN 0454-8124
pISSN 1225-6951

### 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

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