## 3 Results and Discussion
### Hypergraphs  
Figure 9 represents the hypergraph constructed based on a chosen threshold (\(\sigma_{th}=0.5\)) in different forms with colors corresponding to different frequency bands. In particular, the hypergraph is shown as a network in Figure 9a, a star expansion in Figure 9b with connections for each node, and as its incidence matrix in Figure 9c. It is observed that all eight lobes are coupled in the low-frequency ranges of the delta and theta bands, seven lobes (FL, FR, OL, OR, TR, PL, and PR) are coupled for the alpha band, eight lobes (FL, FR, TL, TR, OL, OR, and PL) for the beta band, and six lobes (FL, FR, OL, OR, PL, and PR) for the gamma band.  
The analysis was carried out following the basic properties of hypergraphs. The degrees of the vertices and hyperedges are given in Table 3, where \(deg_{H}(i)=|\varepsilon(i)|\) is the number of vertices incident on \(i\), and the degree of a hyperedge \(deg(h)=|h|\) is the number of vertices it contains.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Vertice** & \(|\varepsilon(i)|\) & **Hyperedge** & \(|\mathbf{e}|\) \\ \hline Frontal Left & 5 & Delta & 8 \\ Frontal Right & 5 & Theta & 8 \\ Occipital Left & 5 & Alpha & 7 \\ Occipital Right & 5 & Beta & 7 \\ Parietal Left & 5 & Gamma & 6 \\ Parietal Right & 4 & & \\ Temporal Left & 3 & & \\ Temporal Right & 4 & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Vertices and hyperedges degrees.  
A star, as mentioned earlier, denotes a pattern where a central vertex--in this case, representing a cerebral lobe--is intricately connected to multiple peripheral vertices, symbolizing different brain frequency ranges. This graphical representation is valuable as it effectively portrays the intricate relationship between a specific cerebral lobe and its involvement across diverse frequency bands. The presence of stars within the graph signifies that the central cerebral lobe exhibits activation across various cognitive conditions or mental states, indicative of its multifunctional nature.  
The stars of each lobe are the following:  
- Left frontal lobe (node 1): Delta, theta, alpha, beta, and gamma.  
- Right frontal lobe (node 2): Delta, theta, alpha, beta, and gamma.  
- Left occipital lobe (node 3): Delta, theta, alpha, beta, and gamma.  
- Right occipital lobe (node 4): Delta, theta, alpha, beta, and gamma.  
- Left parietal lobe (node 5): Delta, theta, alpha, beta, and gamma.  
- Right parietal lobe (node 6): Delta, theta, alpha, and gamma.  
- Left temporal lobe (node 7): Delta, theta, and beta.  
- Right temporal lobe (node 8): Delta, theta, alpha, and beta.  
Although correlations between frequencies have been observed in previous studies [52], and biophysical models have been proposed to explain interactions among different frequency bands, such as theta and gamma [53], further research, similar to the current study, is necessary to elucidate the potential coupling between the mechanisms generating these distinct frequencies.  
Figure 9: Hypergraph representation as (**a**) network, (**b**) star expansion, and (**c**) adjacency matrix. Colors are maintained between hypergraph representations. Delta (red), Theta (blue), Alpha (green), Beta (magenta), Gamma (yellow).