A survey on conjugacy class graphs of groups

IF 0.8 4区数学 Q2 MATHEMATICS Expositiones Mathematicae Pub Date : 2024-06-04 DOI:10.1016/j.exmath.2024.125585

Peter J. Cameron , Firdous Ee Jannat , Rajat Kanti Nath , Reza Sharafdini

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Abstract

There are several graphs defined on groups. Among them we consider graphs whose vertex set consists conjugacy classes of a group $G$ and adjacency is defined by properties of the elements of conjugacy classes. In particular, we consider commuting/nilpotent/solvable conjugacy class graph of $G$ where two distinct conjugacy classes $a^{G}$ and $b^{G}$ are adjacent if there exist some elements $x \in a^{G}$ and $y \in b^{G}$ such that $〈 x, y 〉$ is abelian/nilpotent/solvable. After a section of introductory results and examples, we discuss all the available results on connectedness, graph realization, genus, various spectra and energies of certain induced subgraphs of these graphs. Proofs of the results are not included. However, many open problems for further investigation are stated.

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群的共轭类图概览

有几种图定义在群上。其中，我们考虑顶点集由群 G 的共轭类组成的图，其相邻性由共轭类元素的属性定义。特别是，我们考虑 G 的换元/零能/可解共轭类图，其中如果存在一些元素 x∈aG 和 y∈bG 使得〈x,y〉是无性/零能/可解的〈x,y〉，则两个不同的共轭类 aG 和 bG 相邻。在介绍性结果和示例部分之后，我们讨论了关于这些图的连通性、图实现、属性、各种谱和某些诱导子图的能量的所有可用结果。结果的证明不包括在内。不过，我们也指出了许多有待进一步研究的开放性问题。

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来源期刊

Expositiones Mathematicae 数学-数学

CiteScore

1.30

自引率

0.00%

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审稿时长

40 days

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