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 $(C(S1)∗C(S1)∗⋯∗C(S1)︸|E(Γ)|−times,Δ)$ [in short, $*|E(Γ)|C(S1),Δ$] always acts on a graph C*-algebra for a finite, connected, directed graph Γ in the category introduced by Joardar and Mandal, where |E(Γ)| ≔ number of edges in Γ. 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 $*|E(Γ)|C(S1),Δ$ in the category as mentioned before. More precisely, if a finite, connected, directed graph Γ satisfies the following graph theoretic properties: (i) there does not exist any cycle of length ≥2 (ii) there exists a path of length (|V(Γ)| − 1) which consists all the vertices, where |V(Γ)| ≔ number of vertices in Γ (iii) given any two vertices (may not be distinct) there exists at most one edge joining them, then the universal object coincides with $*|E(Γ)|C(S1),Δ$⁠. Furthermore, we have pointed out a few counter examples whenever the above assumptions are violated.