the spectral representation of a graph filter is independent from the graph

From the spectral representation of a convolutional graph filter $\hat{y}=\sum _{k=0}^{K-1}{h}_k{\mathrm{\Lambda }}^k\hat{x}$, we can see that this representation is completely defined by the polynomial $h(\lambda )=\sum _{k=0}^{K-1}{h}_k{\lambda }^k$ in the spectral domain. ie, the filter is independent from the graph

• We can see from above that the filter frequency response only depends on the coefficients for the filter $h_k$ and the eigenvalues $\mathrm{\Lambda }$ of the shift operator.
• If we consider another graph $\tilde{G}$ with shift operator $\tilde{S}$ and $\sigma (\tilde{S})=\tilde{\mathrm{\Lambda }}$. Then $H(\tilde{S})=\sum _{k=0}^{K-1}{h}_k{\tilde{\mathrm{\Lambda }}}^k$