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

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