Talk by Gökhan Mutlu

On April 21st, 2021, Dr. Gökhan Mutlu (Gazi University) gave a talk about "On the quotient quantum graph with respect to the regular representation" as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.


Show Abstract

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 $|G|$.

Video of the talk

Liza Schonlau | 10.05.2024