Spectrum of the Power Graph of Finite Groups

Ali Reza Ashrafi
University of Kashan, Kashan, I R Iran

Ahmad Gholami
University of Qom, Qom, I R Iran

Zeinab Mehranian
University of Qom, Qom, I R Iran

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Minisymposium: APPLICATIONS OF GROUPS IN GRAPH THEORY

Content: The power graph $\mathcal{P}(G)$ of a group $G$ is the graph with group elements as vertex set and two elements are adjacent if one is a power of the other [1,3]. Chattopadhyay and Panigrahi [2] studied the Laplacian spectrum of the power graph of cyclic group $Z_n$ and the dihedral group $D_{2n}$, $n \geq 2$ is positive integer. They proved that the Laplacian spectrum of $\mathcal{P}(D_{2n})$ is the union of that of $\mathcal{P}(Z_n)$ and $\{ 2n , 1\}$. The aim of this paper is to report our recent results on the spectrum of the power graph of some finite groups. \vskip 3mm \noindent{\bf Keywords:} Power graph, graph spectrum, generalized quaternion, semi-dihedral group. \vskip 3mm \noindent{\bf AMS Subject Classification Number:} $05C25$, $05C50$. \vskip 3mm \noindent{\bf References} \begin{itemize} \item[{[1]}] I. Chakrabarty, S. Ghosh and M. K. Sen, Undirected power graphs of semigroups, Semigroup Forum \textbf{78} (2009) 410--426. \item[{[2]}] S. Chattopadhyaya and P. Panigrahi, On Laplacian spectrum of power graphs of finite cyclic and dihedral groups, Linear Multilinear Algebra \textbf{63}(7) (2015) 1345--1355. \item[{[3]}] A. V. Kelarev and S. J. Quinn, A combinatorial property and power graphs of groups, Contributions to general algebra, 12 (Vienna, 1999), 229--235, Heyn, Klagenfurt (2000). \end{itemize}

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