On the quotient quantum graph with respect to the regular representation

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