On the nilpotent conjugacy class graph of groups
Abstract
The nilpotent conjugacy class graph (or NCC-graph) of a group
is a graph whose vertices are the nontrivial conjugacy classes of
such that two distinct vertices
and
are adjacent if
is nilpotent for some
and
. We discuss on the number of connected components as well as diameter of connected components of these graphs. Also, we consider the induced subgraph
of the NCC-graph with vertices set
, where
, and classify all finite non-nilpotent group
with empty and triangle-free NCC-graphs.
![G](http://212.189.136.205/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![G](http://212.189.136.205/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
![x^G](http://212.189.136.205/plugins/generic/latexRender/cache/62208c36b6e091af9c746598c2411cbf.png)
![y^G](http://212.189.136.205/plugins/generic/latexRender/cache/fc820158504af8272e87e8ad824639bc.png)
![\gen{x',y'}](http://212.189.136.205/plugins/generic/latexRender/cache/9c019739dd2e27cda0d3c0fb1ea976c6.png)
![x'\in x^G](http://212.189.136.205/plugins/generic/latexRender/cache/fa8d1c0790618e4d703a95737bdf80e7.png)
![y'\in y^G](http://212.189.136.205/plugins/generic/latexRender/cache/5193c003df305f70590b4a9b52c7b58e.png)
![\G_n(G)](http://212.189.136.205/plugins/generic/latexRender/cache/deabf07623d4569a9ce9a95b1d7c8cbe.png)
![\{g^G\mid g\in G\setminus\Nil(G)\}](http://212.189.136.205/plugins/generic/latexRender/cache/c73ff1ef9d9df0b04aadea04c79a80ae.png)
![\Nil(G)=\{g\in G\mid\gen{x,g}\text{ is nilpotent for all }x\in G\}](http://212.189.136.205/plugins/generic/latexRender/cache/88af6a39a154491a20205b2e3a19f26e.png)
![G](http://212.189.136.205/plugins/generic/latexRender/cache/dfcf28d0734569a6a693bc8194de62bf.png)
DOI Code:
10.1285/i15900932v37n2p77
Keywords:
Triangle-free; conjugacy class; non-nilpotent group; graph
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