## 3 Methodology
### Platoon Topology Control Based on Graph Theory and Safety Potential Field Theory
In a directed graph, the term degree may refer either to indegree (the number of incoming edges at each vertex) or outdegree (the number of outgoing edges at each vertex). The degree matrix D is used together with the adjacency matrix A to construct the Laplacian matrix \(\mathcal{L}\) of a graph. The expression of the Laplacian matrix \(\mathcal{L}\) can be shown as follows:  
\[\mathcal{L}=D-A. \tag{1}\]  
Assume that the importance of each edge is consistent with the degree matrix, with reference to the definition of the incidence matrix I, the Laplacian matrix \(\mathcal{L}\) can also be expressed in the following variants [33]:  
\[\mathcal{L}=I\cdot D\cdot I^{T}. \tag{2}\]  
After the above analysis, it can be concluded that in our research, the vertexes (\(V\)) and edges (\(E\)) correspond to the controlled CAVs and inter-vehicle communication links for information sharing, respectively. The following theorem is used to offer a stable solution to the platoon formation problem based on graph theory. **Theorem 1**. : _Suppose that the motion control function of the individual vehicle can be expressed by \(f(\cdot)\). \(f(\cdot)\) can be an \(N\)-dimensional vector, and \(f_{i}\) is the value of the function \(f(\cdot)\) at the vertex \(v_{i}\) in the graph. It is assumed that under the CAVs environment, the vehicles in a platoon have a specific communication topology through \(V2V\) (vehicle to vehicle)/\(V2X\) (vehicle to Everything). Changes in the motion of every following vehicle can be directly affected by the state of the leading vehicle in the platoon._  
Then, we have  
1. \(f(\cdot)=\left[ \begin{array}{c}\dot{s}\\ \dot{l}\end{array} \right]\). 2. \(\Delta f(\cdot)=\mathcal{L}\cdot f(\cdot)\). In particular, under certain conditions, the vehicles driving on the road can form a platoon based on graph theory. This means that the longitudinal distance \(s\) and the lateral distance \(l\) can converge to the desired value through the Laplacian matrix \(\mathcal{L}\). **Proof. **_If there is a disturbance in the motion state \(f\) of any vertex \(j\) (\(v_{j}\)), the gain change caused by this disturbance to vertex \(j\) connected to vertex \(i\) (\(v_{j}\)) can be expressed as follows (where, \(j\in N_{i}\), \(N_{i}\) represents the first-order neighborhood node of the vertex \(i\) and considering that the weights of the edges in the graph are the same, equal to 1):_  
\[\Delta f_{i}=\sum_{j\in N_{i}}\big{(}f_{i}-f_{j}\big{)}. \tag{3}\]  
_If edge \(e_{ij}\) has the corresponding weight \(w_{ij}\), then_  
\[\Delta f_{i}=\sum_{j\in N_{i}}w_{ij}\big{(}f_{i}-f_{j}\big{)}. \tag{4}\]  
_When \(w_{ij}=0\), there is no connection between vertex \(i\) and vertex \(j\), that is to say, there is no communication between \(v_{i}\) and \(v_{j}\). Then, Equation (4) can be simplified as_  
\[ \begin{array}{rl}\Delta f_{i}&=\sum\limits_{j\in N}w_{ij}\big{(}f_{i}-f_{j} \big{)}\\ &=\sum\limits_{j\in N}w_{ij}f_{i}-\sum\limits_{j\in N}w_{ij}f_{j}\\ &=d_{i}f_{i}-W_{i}f.\end{array}  \tag{5}\]  
_Consequently, for all of the \(N\) vertexes, we have_  
\[ \begin{array}{rl}\Delta f&=\left(\begin{array}{ccc}d_{1}&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&d_{N}\end{array}\right)f-\left(\begin{array}{c}W_{1}\\ \vdots\\ W_{N}\end{array}\right)f\\ &=diag(d_{i})f-Wf\\ &=(D-W)f\\ &=\mathcal{L}\;f.\end{array}  \tag{6}\]  
_The proof is completed. \(\square\)_  
**Remark 1**. : _According to Theorem 1, in a vehicle platoon composed of CAVs, if the communication topology between CAVs is defined, then a graph based on the topology can be obtained, and the Laplacian matrix of the graph can be applied to achieve platoon formation with the desired distance and speed. In particular, according to whether there is a communication connection between the vehicles, the vehicles connected to each other can correct the difference between the actual distance and the expected distance through real-time communication.