Journal of Guangxi Normal University(Natural Science Edition) ›› 2022, Vol. 40 ›› Issue (6): 116-121.doi: 10.16088/j.issn.1001-6600.2021120903

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Properies of Quasi-Zero-Divisor Graphs of Full Matrix Rings over Zm

ZHAO Shouxiang1,2, TANG Gaohua3*, NAN Jizhu1   

  1. 1. School of Mathematical Sciences, Dalian University of Technology, Dalian Liaoning 116024, China;
    2. Department of Mathematics and Computer Technology, Guilin Normal College, Guilin Guangxi 541199, China;
    3. School of Sciences, Beibu Gulf University, Qinzhou Guangxi 535011, China
  • Received:2021-12-09 Revised:2022-02-24 Online:2022-11-25 Published:2023-01-17

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Abstract: In the past two decades, the zero-divisor graphs that combine ring theory and graph theory have been a hot spot in mathematical research. Many scholars have defined a variety of graphs on the ring according to certain relationships, which are used to study the relationship between the properties of the rings and the properties of the graphs. This paper studies the properties of the quasi-zero divisor graphs of the full matrix rings over the residual class rings. The necessary and sufficient conditions for the matrix to be a vertex are shown in the quasi-zero divisor graphs of the full matrix rings over the residual class rings, and the necessary and sufficient conditions for any two vertices are shown in the quasi-zero divisor graphs of the full matrix rings that the distance between them is equal to 1,2,3. Finally it is proved that the quasi-zero divisor graphs of the full matrix rings over the two residual class rings are isomorphic if and only if the ground ring of the full matrix rings are isomorphic, and the order of the full matrix rings are the same.

Key words: zero divisor graph, quasi-zero-divisor graph, full matrix ring, residue class ring, diameter of graphs

CLC Number: 

  • O153.3
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