  Quantum symmetry of graph $C^{*}$-algebras has been studied, under the
consideration of different formulations, in the past few years. It is already
known that the compact quantum group $(\underbrace{C(S^{1})*C(S^{1})*\cdots
*C(S^{1})}_{|E(\Gamma)|-times}, \Delta) $ always acts on a graph $C^*$-algebra
for a finite, connected, directed graph $\Gamma$ in the category introduced by
Joardar and Mandal, where $|E(\Gamma)|:=$ number of edges in $\Gamma$. In this
article, we show that for a certain class of graphs including Toeplitz algebra,
quantum odd sphere, matrix algebra etc. the quantum symmetry of their
associated graph $C^*$-algebras remains $(\underbrace{C(S^{1})*C(S^{1})*\cdots
*C(S^{1})}_{|E(\Gamma)|-times}, \Delta) $ in the category as mentioned before.
More precisely, if a finite, connected, directed graph $\Gamma$ satisfies the
following graph theoretic properties : (i) there does not exist any cycle of
length $\geq$ 2 (ii) there exists a path of length $(|V(\Gamma)|-1)$ which
consists all the vertices, where $|V(\Gamma)|:=$ number of vertices in $\Gamma$
(iii) given any two vertices (may not be distinct) there exists at most one
edge joining them, then the universal object coincides with
$(\underbrace{C(S^{1})*C(S^{1})*\cdots *C(S^{1})}_{|E(\Gamma)|-times}, \Delta)
$. Furthermore, we have pointed out a few counter examples whenever the above
assumptions are violated.
