On the quotient quantum graph with respect to the regular representation

Mutlu G.

Communications on Pure and Applied Analysis, cilt.20, sa.2, ss.885-902, 2021 (SCI-Expanded) identifier identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 20 Sayı: 2
  • Basım Tarihi: 2021
  • Doi Numarası: 10.3934/cpaa.2020295
  • Dergi Adı: Communications on Pure and Applied Analysis
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Applied Science & Technology Source, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.885-902
  • Anahtar Kelimeler: Isospectrality, quotient quantum graphs, regular representation, per-mutation representation, quantum graphs, ONE HEAR, SHAPE, DOMAINS
  • Gazi Üniversitesi Adresli: Evet

Özet

© 2020 American Institute of Mathematical Sciences. All rights reserved.Given a quantum graph Γ, a finite symmetry group G acting on it and a representation R of G, the quotient quantum graph Γ/R is described and constructed in the literature [1, 2, 18]. In particular, it was shown that the quotient graph Γ/CG is isospectral to Γ by using representation theory where CG denotes the regular representation of G [18]. Further, it was conjectured that Γ can be obtained as a quotient Γ/CG [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 Γ and a finite symmetry group G acting on it, the quotient quantum graph Γ/CG is not only isospectral but rather identical to Γ for a particular choice of a basis for CG. 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|.