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

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.