## 3 Methodology
### Platoon Topology Control Based on Graph Theory and Safety Potential Field Theory  
Figure 1 depicts the information transfer topology diagram in a vehicle platoon, including a leading vehicle (noted as the platoon leader and marked in color) and other following vehicles (noted as the follower vehicles, and marked sequentially from 1 to _i_). Communication technologies such as vehicle to vehicle (V2V)/vehicle to infrastructure (V2I) can be applied to realize the construction of the information flow topology, which can help CAVs to obtain the motion status information of surrounding vehicles. The arrows in Figure 1 represent the topological structure of information transmission between the vehicles in the platoon. These CAVs are capable of using this information to correct the distance (longitudinal and lateral) to surrounding vehicles and ultimately achieve formation control over the platoon. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Strategy**} & \multirow{2}{*}{**References**} & \multicolumn{4}{c}{**Realizable Functions**} \\ \cline{3-5}  & & **Formation** & **Formation** & **Formation** & **Formation** \\  & & **Shape** & **Trajectory** & **Reconfiguration** & **Task Assignment** \\  & & **Generation** & **Tracking** & **Reconfiguration** & **Task Assignment** \\ \hline Virtual structure & [17; 18] & & \(\surd\) & & \\ Behavior-based & [19; 20] & \(\surd\) & \(\surd\) & & \(\surd\) \\ Leader–follower & [21; 22] & & \(\surd\) & & \\ Graph-based & [23; 24] & \(\surd\) & \(\surd\) & \(\surd\) & \(\surd\) \\ Artificial potential field & [25; 26] & \(\surd\) & \(\surd\) & \(\surd\) & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of different platoon formation control strategies. Figure 1: The information transfer topology diagram in a vehicle platoon. The vehicle platoon in Figure 1 can be described by graph-based theory [31; 32]. In graph theory, a directed graph has the expression \(G=\left(V^{*},\,E\right)\), where \(V^{*}\) and \(E\) represent the vertex set (\(V^{*}=\left\{v_{i}^{*},i=1\cdots n\right\}\)) and the edge set (\(E\subseteq V^{*}\times V^{*},E=\left\{e_{k},k=1\cdots n\right\}\)) in the directed graph, respectively. Every vertex represents an individual vehicle (\(v_{i}^{*}\)) and each edge represents a potential communication link (\(e_{k}\)). The directed edge implies the direction of communication transmission. For example, if the communication is transmitted from \(v_{i}^{*}\) to \(v_{j}^{*}\), then \(v_{i}^{*}\) is defined as the parent node and \(v_{j}^{*}\) is defined as the child node. Then, the directed edge has the expression \(e_{k}=\left(v_{i}^{*},v_{j}^{*}\right)\). Otherwise, the expression of \(e_{k}\) will be rewritten as \(e_{k}=\left(v_{j}^{*},v_{i}^{*}\right)\) if the communication is transmitted from \(v_{i}^{*}\) to \(v_{i}^{*}\). The adjacency matrix \(A=\left[a_{ij}\right]\) is a commonly used representation of a graph structure. It uses a digital square matrix to record whether there are edges connected between the vertexes. In a directed graph, adjacent matrices are not necessarily symmetric. Since the element in the \(i\)-th row and the \(j\)-th column of the adjacent matrix is 1, it means that there is an edge from vertex \(v_{i}^{*}\) to vertex \(v_{j}^{*}\), and there is not necessarily an edge from vertex \(v_{j}^{*}\) to vertex \(v_{i}^{*}\) at this time. Meanwhile, the incidence matrix \(I=\left[I_{ij}\right]\) is a matrix that shows the relationship between two classes of objects: the vertex (\(v^{*}\)) and the edge (\(e\)). The values of the elements in the incidence matrix I should follow the rules below: (1) if the edge \(e\) is directed to the vertex \(v\), then the value \(I_{ij}(v^{*},e)=-1\); (2) if the edge \(e\) is directed from the vertex \(v^{*}\), then the value \(I_{ij}(v^{*},e)=1\); (3) otherwise, the value \(I_{ij}(v^{*},e)=0\) is used. The degree matrix \(D=\left[d_{i}\right]\) contains information about the degree of each vertex.