On the diameters of commuting graphs
On the diameters of commuting graphs
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DOI：
10.1016/j.laa.2006.01.029
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发表时间：
2006-10
影响因子：
1.1
通讯作者：
S. Akbari;A. Mohammadian;H. Radjavi;P. Raja
中科院分区：
文献类型：
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作者：
S. Akbari;A. Mohammadian;H. Radjavi;P. Raja
表示为Γ(R)的环R的交换图是这样一个图，其顶点均为R的非中心元素，且两个不同的顶点x和y相邻当且仅当xy=yx。设D是除法环，n大于或等于3。本文研究了Γ（Mn(D)）的直径，并确定了Γ（Mn(D)）的一些诱导子图的直径，例如Mn(D)中所有非标量不可逆、幂零、幂等和对合矩阵集合上的诱导子图。对于每个域F，如果Γ（Mn(F)）是连通图，则diamΓ(Mn(F))≥6。我们推测，如果Γ（Mn(F)）是连通图，则diamΓ(Mn(F))≥5。我们证明了如果F是代数闭域或n是素数，Γ（Mn(F)）是连通图，则diamΓ(Mn(F))=4。最后，我们给出了一些在幂等对结构上的应用。
The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all non-central elements of R and two distinct vertices x and y are adjacent if and only if xy=yx. Let D be a division ring and n⩾3. In this paper we investigate the diameters of Γ(Mn(D)) and determine the diameters of some induced subgraphs of Γ(Mn(D)), such as the induced subgraphs on the set of all non-scalar non-invertible, nilpotent, idempotent, and involution matrices in Mn(D). For every field F, it is shown that if Γ(Mn(F)) is a connected graph, then diamΓ(Mn(F))⩽6. We conjecture that if Γ(Mn(F)) is a connected graph, then diamΓ(Mn(F))⩽5. We show that if F is an algebraically closed field or n is a prime number and Γ(Mn(F)) is a connected graph, then diamΓ(Mn(F))=4. Finally, we present some applications to the structure of pairs of idempotents which may prove of independent interest.