On the quotient quantum graph with respect to the regular representation

Given a quantum graph \begin{document}$ \Gamma $\end{document} , a finite symmetry group \begin{document}$ G $\end{document} acting on it and a representation \begin{document}$ R $\end{document} of \begin{document}$ G $\end{document} , the quotient quantum graph \begin{document}$ \Gamma /R $\end{document} is described and constructed in the literature [ 1 , 2 , 18 ]. In particular, it was shown that the quotient graph \begin{document}$ \Gamma/\mathbb{C}G $\end{document} is isospectral to \begin{document}$ \Gamma $\end{document} by using representation theory where \begin{document}$ \mathbb{C}G $\end{document} denotes the regular representation of \begin{document}$ G $\end{document} [ 18 ]. Further, it was conjectured that \begin{document}$ \Gamma $\end{document} can be obtained as a quotient \begin{document}$ \Gamma/\mathbb{C}G $\end{document} [ 18 ]. However, proving this by construction of the quotient quantum graphs has remained as an open problem. In this paper, we solve this problem by proving by construction that for a quantum graph \begin{document}$ \Gamma $\end{document} and a finite symmetry group \begin{document}$ G $\end{document} acting on it, the quotient quantum graph \begin{document}$ \Gamma / \mathbb{C}G $\end{document} is not only isospectral but rather identical to \begin{document}$ \Gamma $\end{document} for a particular choice of a basis for \begin{document}$ \mathbb{C}G $\end{document} . Furthermore, we prove that, this result holds for an arbitrary permutation representation of \begin{document}$ G $\end{document} with degree \begin{document}$ |G| $\end{document} , whereas it doesn't hold for a permutation representation of \begin{document}$ G $\end{document} with degree greater than \begin{document}$ |G|. $\end{document}