Quantum Symmetries of Graph C*-algebras Having Maximal Permutational Symmetry

Quantum symmetry of a graph $C^{*}$-algebra $C^{*}(\Gamma)$ corresponding to a finite graph $\Gamma$ has been explored by several mathematicians within different categories in the past few years. In this article, we establish that there are exactly three families of compact matrix quantum groups, containing the symmetric group on the set of edges of the underlying graph $\Gamma$, that can be achieved as the quantum symmetries of graph $C^*$-algebras in the category introduced by Joardar and Mandal. Moreover, we demonstrate that there does not exist any graph $C^*$-algebra associated with a finite graph $\Gamma$ without isolated vertices having $A_{u^t}(F^{\Gamma})$ as the quantum automorphism group of $C^*(\Gamma)$ for a non-scalar matrix $F^{\Gamma}$.