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On the nilpotent conjugacy class graph of groups


Abstract


The nilpotent conjugacy class graph (or NCC-graph) of a group G is a graph whose vertices are the nontrivial conjugacy classes of G such that two distinct vertices x^G and y^G are adjacent if \gen{x',y'} is nilpotent for some x'\in x^G and y'\in y^G. We discuss on the number of connected components as well as diameter of connected components of these graphs. Also, we consider the induced subgraph \G_n(G) of the NCC-graph with vertices set \{g^G\mid g\in G\setminus\Nil(G)\}, where \Nil(G)=\{g\in G\mid\gen{x,g}\text{ is nilpotent for all }x\in G\}, and classify all finite non-nilpotent group G with empty and triangle-free NCC-graphs.

DOI Code: 10.1285/i15900932v37n2p77

Keywords: Triangle-free; conjugacy class; non-nilpotent group; graph

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