The paper by S. Alexander and R. Orbach calculates the density of states on fractals, considering the scaling properties of both volume and connectivity. They use a Green's function method that relates the problem to diffusion. The key finding is that proper mode counting requires a reciprocal space with a new dimensionality, called the "fracton dimensionality" ($\overline{d}$), given by $\overline{d} = 2 \overline{d} (2 + \overline{\delta})$. Here, $\overline{d}$ is the effective dimensionality, and $\overline{\delta}$ is the exponent characterizing the variation of the diffusion constant with distance. For percolation clusters, they find $\overline{d} = 4/3$, independent of the Euclidean dimensionality $d$. The paper discusses the crossover behavior to normal behavior at low frequencies for finite fractals and percolation above the threshold $p_c$. They also explore the relevance of their predictions to experimental results on proteins. The authors provide examples for one-dimensional chains, the triangular Sierpinski gasket, and critical percolation networks, showing how the fracton dimensionality affects the eigenvalue density of states. They conclude with a discussion on the experimental relevance of their findings, particularly in the context of protein vibrational density of states.The paper by S. Alexander and R. Orbach calculates the density of states on fractals, considering the scaling properties of both volume and connectivity. They use a Green's function method that relates the problem to diffusion. The key finding is that proper mode counting requires a reciprocal space with a new dimensionality, called the "fracton dimensionality" ($\overline{d}$), given by $\overline{d} = 2 \overline{d} (2 + \overline{\delta})$. Here, $\overline{d}$ is the effective dimensionality, and $\overline{\delta}$ is the exponent characterizing the variation of the diffusion constant with distance. For percolation clusters, they find $\overline{d} = 4/3$, independent of the Euclidean dimensionality $d$. The paper discusses the crossover behavior to normal behavior at low frequencies for finite fractals and percolation above the threshold $p_c$. They also explore the relevance of their predictions to experimental results on proteins. The authors provide examples for one-dimensional chains, the triangular Sierpinski gasket, and critical percolation networks, showing how the fracton dimensionality affects the eigenvalue density of states. They conclude with a discussion on the experimental relevance of their findings, particularly in the context of protein vibrational density of states.