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

#### Abstract

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.

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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