Spectrum and $L-$Spectrum of the Power Graph and its Main Supergraph for certain Finite Groups

Asma Hamzeh, Ali Reza Ashrafi


Let $G$ be a finite group. The power graph $\mathcal{P}(G)$
and its main supergraph  $\mathcal{S}(G)$ are two simple graphs with the same vertex set $G$. Two elements $x,y \in G$ are adjacent in the
power graph if and only if one is a power of the other. They are
joined in $\mathcal{S}(G)$ if and only if $o(x) |
o(y)$ or $o(y) | o(x)$. The aim of this paper is to compute the characteristic polynomial of these graph for  certain finite groups. As a consequence, the spectrum and Laplacian spectrum  of these graphs for dihedral, semi-dihedral, cyclic and dicyclic groups were computed.

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