  In 2009, T. Banica and R. Speicher introduced the orthogonal easy quantum
group which can be completely determined using the combinatorics of set
partitions. Later, P. Tarrago and M. Weber extended this formulation to another
special class of compact matrix quantum groups, known as unitary easy quantum
groups, which are quantum subgroups of the free unitary quantum group ($U_n^+$)
containing $S_n$. On the other hand, the quantum symmetry of graph
$C^{*}$-algebras 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 unitary easy quantum groups that can be achieved as
the quantum symmetry of graph $C^*$-algebra $C^*(\Gamma)$ associated with a
finite, connected, directed graph $\Gamma$ in the category introduced by
Joardar and Mandal. Moreover, we demonstrate that there does not exist any
graph $C^*$-algebra associated with a finite, connected, directed graph
$\Gamma$ having $A_{U^t}(F^{\Gamma})$ as the quantum automorphism group of
$C^*(\Gamma)$ for non-scalar matrix $F^{\Gamma}$.
