On quantum symmetries of graphs

• Date: 2026
• Main Authors: Ostrovska, O., Ostrovskyi, V., Turowska, L., Островська, Ольга, Островський, Василь, Туровська, Людмила, Островский, Василий
• Format: Article
• Language: Ukrainian
• Published: Institute of Mathematics, NAS of Ukraine 2026
• Online Access: https://umj.imath.kiev.ua/index.php/umj/article/view/9774
• Journal Title: Ukrains’kyi Matematychnyi Zhurnal

UDC 519.17, 517.98, 519.8 Let $G$ be a simple finite graph and let $\mathcal U_G$ be the corresponding quantum graph. We study the game algebra $C(\mathrm{Qut}(\mathcal U_G))$ of the quantum automorphisms of $\mathcal U_G.$ It is shown that, for the complete graph $K_n,$ the algebra $C(\mathrm{Qut}(\mathcal U_{K_n}))$ is not commutative even for all $n\geq 3,$ unlike $C(\mathrm{Qut}(K_n))=C(S_n^+).$ Moreover, we prove that, for any graph $G$ with $|V(G)|\geq 3,$ the quantum graph $\mathcal U_G$ admits nonlocal symmetry, which means that there exists a perfect quantum no-signaling correlation for the quantum automorphism game for $\mathcal U_G,$ which is not local.