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Volume 28, Issue 4
On Commuting Graph of Group Ring $Z_nS_3$

Yanyan Gao, Gaohua Tang & Jianlong Chen

Commun. Math. Res., 28 (2012), pp. 313-323.

Published online: 2021-05

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  • Abstract

The commuting graph of an arbitrary ring $R$, denoted by $Γ(R)$, is a graph whose vertices are all non-central elements of $R$, and two distinct vertices $a$ and $b$ are adjacent if and only if $ab = ba$. In this paper, we investigate the connectivity and the diameter of $Γ(Z_n S_3)$. We show that $Γ(Z_n S_3)$ is connected if and only if $n$ is not a prime number. If $Γ(Z_n S_3)$ is connected then diam $(Γ(Z_n S_3)) = 3$, while if $Γ(Z_n S_3)$ is disconnected then every connected component of $Γ(Z_n S_3)$ must be a complete graph with same size, and we completely determine the vertice set of every connected component.

  • Keywords

group ring, commuting graph, connected component, diameter of a graph.

  • AMS Subject Headings

16S34, 20C05, 05C12, 05C40

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{CMR-28-313, author = {Gao , YanyanTang , Gaohua and Chen , Jianlong}, title = {On Commuting Graph of Group Ring $Z_nS_3$}, journal = {Communications in Mathematical Research }, year = {2021}, volume = {28}, number = {4}, pages = {313--323}, abstract = {

The commuting graph of an arbitrary ring $R$, denoted by $Γ(R)$, is a graph whose vertices are all non-central elements of $R$, and two distinct vertices $a$ and $b$ are adjacent if and only if $ab = ba$. In this paper, we investigate the connectivity and the diameter of $Γ(Z_n S_3)$. We show that $Γ(Z_n S_3)$ is connected if and only if $n$ is not a prime number. If $Γ(Z_n S_3)$ is connected then diam $(Γ(Z_n S_3)) = 3$, while if $Γ(Z_n S_3)$ is disconnected then every connected component of $Γ(Z_n S_3)$ must be a complete graph with same size, and we completely determine the vertice set of every connected component.

}, issn = {2707-8523}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/cmr/19034.html} }
TY - JOUR T1 - On Commuting Graph of Group Ring $Z_nS_3$ AU - Gao , Yanyan AU - Tang , Gaohua AU - Chen , Jianlong JO - Communications in Mathematical Research VL - 4 SP - 313 EP - 323 PY - 2021 DA - 2021/05 SN - 28 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/cmr/19034.html KW - group ring, commuting graph, connected component, diameter of a graph. AB -

The commuting graph of an arbitrary ring $R$, denoted by $Γ(R)$, is a graph whose vertices are all non-central elements of $R$, and two distinct vertices $a$ and $b$ are adjacent if and only if $ab = ba$. In this paper, we investigate the connectivity and the diameter of $Γ(Z_n S_3)$. We show that $Γ(Z_n S_3)$ is connected if and only if $n$ is not a prime number. If $Γ(Z_n S_3)$ is connected then diam $(Γ(Z_n S_3)) = 3$, while if $Γ(Z_n S_3)$ is disconnected then every connected component of $Γ(Z_n S_3)$ must be a complete graph with same size, and we completely determine the vertice set of every connected component.

Gao , YanyanTang , Gaohua and Chen , Jianlong. (2021). On Commuting Graph of Group Ring $Z_nS_3$. Communications in Mathematical Research . 28 (4). 313-323. doi:
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