The commuting graph of an arbitrary ring R, denoted by Г（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 inv...The commuting graph of an arbitrary ring R, denoted by Г（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 Г（ZnS3）. We show that Г（ZnS3） is connected if and only if n is not a prime number. If Г（ZnS3） is connected then diam（Г（ZnS3）） = 3, while ifГ（ZnS3） is disconnected then every connected component of Г（ZnS3） must be a complete graph with same size, and we completely determine the vertice set of every connected component.展开更多
For a group G and a non-empty subsetΩof G,the commuting graph C(G,Ω)ofΩis a graph whose vertex set isΩand any two vertices are adjacent if and only if they commute in G.Define T4n=(a,b|a^(2)n=b^(4)=1,an=b2,b^(−1)a...For a group G and a non-empty subsetΩof G,the commuting graph C(G,Ω)ofΩis a graph whose vertex set isΩand any two vertices are adjacent if and only if they commute in G.Define T4n=(a,b|a^(2)n=b^(4)=1,an=b2,b^(−1)ab=a^(−1)),the dicyclic group of order 4n(n≥3),which is also known as the generalized quaternion group.We mainly investigate the properties and metric dimension of the commuting graphs on the dicyclic group T4n.展开更多
In this paper, we prove that the n-simple braid divisible by the generators xi for all 2 ≤ i ≤n - 2 has trivial simple centralizer. Consequently, the commuting graph defined on the set of simple braids is disconnect...In this paper, we prove that the n-simple braid divisible by the generators xi for all 2 ≤ i ≤n - 2 has trivial simple centralizer. Consequently, the commuting graph defined on the set of simple braids is disconnected. We also prove that the graph has one major component.展开更多
基金The NSF(10971024)of Chinathe Specialized Research Fund(200802860024)for the Doctoral Program of Higher Educationthe NSF(BK2010393)of Jiangsu Province
文摘The commuting graph of an arbitrary ring R, denoted by Г（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 Г（ZnS3）. We show that Г（ZnS3） is connected if and only if n is not a prime number. If Г（ZnS3） is connected then diam（Г（ZnS3）） = 3, while ifГ（ZnS3） is disconnected then every connected component of Г（ZnS3） must be a complete graph with same size, and we completely determine the vertice set of every connected component.
基金This work was supported by NSFC Grant 11871206Natural Science Foundation of Hunan Province(No.2020JJ4233,No.2020JJ5096).
文摘For a group G and a non-empty subsetΩof G,the commuting graph C(G,Ω)ofΩis a graph whose vertex set isΩand any two vertices are adjacent if and only if they commute in G.Define T4n=(a,b|a^(2)n=b^(4)=1,an=b2,b^(−1)ab=a^(−1)),the dicyclic group of order 4n(n≥3),which is also known as the generalized quaternion group.We mainly investigate the properties and metric dimension of the commuting graphs on the dicyclic group T4n.
文摘In this paper, we prove that the n-simple braid divisible by the generators xi for all 2 ≤ i ≤n - 2 has trivial simple centralizer. Consequently, the commuting graph defined on the set of simple braids is disconnected. We also prove that the graph has one major component.
