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Quantum symmetry of the graph $C^{*}$-algebras has been studied by severalauthors, under the consideration of different formulations, in recent times. Itis 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$\Gamma=\{V(\Gamma), E(\Gamma), r, s\}$ in the category introduced by Joardarand Mandal. In this article, we have shown that for a certain class of graphsincluding Toeplitz algebra, quantum odd sphere, matrix algebra etc. the quantumsymmetry of their associated graph $C^*$-algebras remains$(\underbrace{C(S^{1})*\cdots * C(S^{1})}_{|E(\Gamma)|-times}, \Delta) $ in thecategory mentioned before. More precisely, if a finite, connected, directedgraph $\Gamma=\{V(\Gamma),E(\Gamma),r,s\}$ satisfies the following graphtheoretic 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 thevertices (iii) given any two vertices (may not be distinct) there exists atmost 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 few counter examples whenever the aboveassumptions are violated.

Rigidity on Quantum Symmetry for a Certain Class of Graph C*-algebras