Quantum symmetries of graph C∗-algebras having maximal permutational symmetry

Quantum symmetry of a graph [Formula: see text]-algebra [Formula: see text] corresponding to a finite graph [Formula: see text] has been explored by several mathematicians within different categories in the past few years. In this paper, 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 [Formula: see text], that can be achieved as the quantum symmetries of graph [Formula: see text]-algebras in the category introduced by Joardar and Mandal. Moreover, we demonstrate that there does not exist any graph [Formula: see text]-algebra associated with a finite graph [Formula: see text] without isolated vertices having [Formula: see text] as the quantum automorphism group of [Formula: see text] for a non-scalar matrix [Formula: see text].