Isometric actions of compact quantum groups on graph C*-algebras

For a strongly connected directed graph $\mathrm{\Lambda }$, it is shown that the quantum automorphism group of $\mathrm{\Lambda }$ acts isometrically in the sense of D. Goswami et al. on the spectral triple $(C^{\ast }(\mathrm{\Lambda }),L^2({\mathrm{\Lambda }}^{\mathrm{\infty }},M),D)$ constructed by Farsi et al… It is also shown that the natural action of $U_n^{+}$ is not isometric on the Cuntz algebra ${\mathcal{O}}_n$.