Talk by Gökhan Mutlu

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, 3]. Different choices for the fundamental domain of the action of $G$ on $\Gamma $ and for the basis of $R$ yield different quotient graphs $\Gamma /R$ which are all isospectral to each other. In particular, it was proven that the quotient graph $\Gamma / \mathbb{C} G$ is isospectral to $\Gamma$ where $\mathbb{C} G$ denotes the regular representation of $G$ [3]. Choosing different fundamental domains for the action of $G$ or different basis of $\mathbb{C} G$ will yield new quantum graphs which are isospectral to $\Gamma$. This provides an extremely useful way to create isospectral quantum graphs to a given quantum graph. It was conjectured that $\Gamma$ is a $\Gamma / \mathbb{C} G$ graph i.e. $\Gamma$ can be obtained as a quotient $\Gamma / \mathbb{C} G$ for a particular choice of a basis for $\mathbb{C} G$ [3]. However, proving this by construction of the quotient quantum graphs has remained as an open problem.

In this study, we construct the quotient quantum graph $\Gamma / \mathbb{C} G$ by choosing G as a basis for $\mathbb{C} G$ and show that the resulting graph is identical to $\Gamma$. Moreover, we prove a more general result that $\Gamma$ can be obtained as a quotient $\Gamma / \rho$ where $\rho$ is an arbitrary permutation representation of $G$ with degree $|G|$. We prove that if one constructs the quotient graph $\Gamma / \rho$ by choosing the standard basis of $\mathbb{C}^{|G|}$, one gets $\Gamma$ where $\rho$ is an arbitrary permutation representation of $G$ with degree $|G|$. We also show by a counterexample that this does not hold for a permutation representation of $G$ with degree greater than $|G|$.