Multiscale Spectral Graphs for Structural Analysis

In the past few years, we have developed a multiscale spectral graph method such as generalized GNM and generalized ANM [33 , 34 ], to create a family of spectral graphs with different characteristic length scales for a given dataset. Similarly, in our persistent spectral theory, we can construct a family of spectral graphs induced by a filtration parameter. Moreover, we can sum over all the multiscale spectral graphs as an accumulated spectral graph. Specifically, a family of ${\mathcal{L}}_0^{r+0}$ matrices, as well as the accumulated combinatorial Laplacian matrices, can be generated via the filtration. By analyzing the persistent spectra of these matrices, the topological invariants and geometric shapes can be revealed from the given input point-cloud data. The spectra of ${\mathcal{L}}_0^{r+0}$ , ${\widehat{\mathcal{L}}}_0^{r+0}$ , and ${\check{\mathcal{L}}}_0^{r+0}$ mentioned above carry similar information on how the topological structures of a graph are changed during the filtration. Benzene molecule (C₆H₆), a typical aromatic hydrocarbon which is composed of six carbon atoms bonded in a planar regular hexagon ring with one hydrogen joined with each carbon atom. It provides a good example to demonstrate the proposed PST. Figure 4 illustrates the filtration of the benzene molecule. Here, we label 6 hydrogen atoms by H₁, H₂, H₃, H₄, H₅, and H₆, and the carbon adjacent to the labeled hydrogen atoms are labeled by C₁, C₂, C₃, C₄, C₅, and C₆, respectively. Figure 5(b) depicts that when the radius of the solid sphere reaches 0.54 Å, each carbon atom in the benzene ring is overlapped with its joined hydrogen atom, resulting in the reduction of ${\beta }_0^{r+0}$ to 6. Moreover, once the radius of solid spheres is larger than 0.70 Å, all the atoms in the benzene molecule will connect and constitute a single component which gives rise ${\beta }_0^{r+0}=1$ . Furthermore, we can deduce that the C-C bond length of the benzene ring is about 1.40 Å, and the C-H bond length is around 1.08 Å, which are the real bond lengths in benzene molecule. Figure 5(c) shows that a 1-dimensional hole (1-cycle) is born when the filtration parameter r increase to 0.70 Å and dead when r = 1.21 Å. In Figures 5(b) and 5(c), it can be seen that variants of 0-persistent 0-combinatorial Laplacian and 1 -combinatorial Laplacian matrices based on filtration give us the identical ${\beta }_0^{r+0}$ and ${\beta }_1^{r+0}$ information respectively.
The C-C bond length of benzene is 1.39 Å, and the C-H bond length is 1.09 Å. Due to the perfect hexagon structure of the benzene ring, we can calculate all of the distances between atoms. The shortest and longest distances between carbons and the hydrogen atoms are 1.09 Å and 3.87 Å. In Figure 5(a), a total of 10 changes of ${\left({\tilde{\lambda }}_2\right)}_0^{r+0}$ values is observed at various radii. Table 4 lists all the distances between atoms and the values of radii when the changes of ${\left({\tilde{\lambda }}_2\right)}_0^{r+0}$ occur. It can be seen that the distance between atoms approximately equals twice of the radius value when a jump of ${\left({\tilde{\lambda }}_2\right)}_0^{r+0}$ occurs. Therefore, we can detect all the possible distances between atoms with the nonzero spectral information. Moreover, in Figure 5(b), the values of the smallest nonzero eigenvalues of ${\mathcal{L}}_0^{r+0}$ , ${\widehat{\mathcal{L}}}_0^{r+0}$ , and ${\check{\mathcal{L}}}_0^{r+0}$ change concurrently.