Geodesic
Spectral Properties of $NC$-Graphs
Kragujevac Journal of Mathematics, Tome 43 (2019) no. 4, p. 523

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Let $G$ be a non-abelian group and $Z(G)$ be the center of $G$. The non-commuting graph ($NC$-graph) $\Gamma(G)$ of the group $G$ is a graph with the vertex set $G\setminus Z(G)$ and two distinct vertices $x$ and $y$ are adjacent whenever $xy \neq yx$. The aim of this paper is to prove that for given group $G$, $\frac{G}{Z(G)}\cong \Bbb{Z}_p×\Bbb{Z}_p$ if and only if $\Gamma(G)$ is a regular ($p+1$)-partite graph. Also we consider the isomorphism of the non-commuting graph with some special graphs.
Classification : 05E15 05C25, 20D99
Keywords: Non-commuting graph, centralizer, $p$-groups 
@article{KJM_2019_43_4_a2,
     author = {M. Ghorbani and Z. Gharavi-Alkhansari},
     title = {Spectral {Properties} of $NC${-Graphs}},
     journal = {Kragujevac Journal of Mathematics},
     pages = {523 },
     publisher = {mathdoc},
     volume = {43},
     number = {4},
     year = {2019},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/KJM_2019_43_4_a2/}
}
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M. Ghorbani; Z. Gharavi-Alkhansari. Spectral Properties of $NC$-Graphs. Kragujevac Journal of Mathematics, Tome 43 (2019) no. 4, p. 523 . http://geodesic.mathdoc.fr/item/KJM_2019_43_4_a2/