## 2 Problem Statement  
### Preliminaries  
In this subsection, the model of the communication topology, definition of the Dixon resultant, and CTCR paradigm are demonstrated. Firstly, the model of the communication topology is given. The platoon consists of one leader and \(N\) followers. In our work, the commonly used PF topology is chosen as the communication topology, as illustrated in Figure 2. The red arrows denote the transmission of information (i.e., position and velocity) through onboard sensors like radars, while the blue dashed arrows represent the transmission of information (i.e., acceleration) through  
Figure 1: Architecture of the stability analysis of the vehicular platoon. wireless communication networks. Furthermore, the integrated sensors are used to detect the position and velocity information of the preceding vehicle simultaneously, ensuring an identical sensing time for the above types of information [30]. The adjacent matrix of the PF topology is defined as  
\[\mathcal{A}=\left[a_{ij}\right]_{N\times N}= \begin{bmatrix}0&0&&\\ 1&0&0&&\\ &\ddots&\ddots&\ddots&\\ &&1&0&0\\ &&&1&0\end{bmatrix} \]  
where \(a_{ij}=1\) represents that the follower \(i\) can receive the information from the follower \(j\); otherwise, \(a_{ij}=0\). \(i=1,2,\cdots,N\), \(j=1,2,\cdots,N\). The indegree matrix is characterized by \(\mathcal{D}=\text{diag}\{\text{deg}_{1},\text{deg}_{2},\cdots,\text{deg}_{N}\}\), \(\text{deg}_{i}=\sum_{j=1}^{N}a_{ij}\). According to the two above-mentioned matrices, the Laplacian matrix is modeled by \(\mathcal{L}=\mathcal{D}-\mathcal{A}\). Then, the pinning matrix \(\mathcal{P}\), which characterizes the information flow from the leader to the followers, is defined as  
\[\mathcal{P}=\left[p_{ii}\right]_{N\times N}=\text{diag}\{1,0,\cdots,0\}\]  
where \(p_{ii}=1\) represents that the follower \(i\) can receive the information from the leader; otherwise, \(p_{ii}=0\). We utilize the augmented Laplacian matrix \(\widetilde{\mathcal{L}}=\mathcal{L}+\mathcal{P}\) to characterize the communication topology. For PF topology, every eigenvalue of \(\widetilde{\mathcal{L}}\) is 1. Secondly, the definition of the Dixon resultant is given. The Dixon resultant is a peer methodology to reveal the sufficient and necessary conditions of the nontrivial common solutions of the polynomial equations [31]. It outperforms the commonly used resultant formulation (i.e., Sylvester resultant) in computational efficiency [21]. The determinant of the Dixon resultant is equal to two polynomials, which can be used to implement the first step of the CTCR paradigm. The two polynomials are denoted as \(\psi_{1}(x)\) and \(\psi_{2}(x)\), where \(x\) is a variable. Then, the Dixon polynomial is given by  
\[\mu(x,\alpha)=\frac{1}{(x-\alpha)}\begin{vmatrix}\psi_{1}(x)&\psi_{2}(x)\\ \psi_{1}(\alpha)&\psi_{2}(\alpha)\end  
If there exists a nontrivial solution for \(\psi_{1}(x)\) and \(\psi_{2}(x)\), \(D(\psi_{1}(x),\psi_{2}(x))\) must be singular [31].  
\[\det[D(\psi_{1}(x),\psi_{2}(x))]=0 \tag{3}\]  
The determinant in (3) is called the Dixon resultant. At last, some definitions of the CTCR paradigm are provided. The CTCR paradigm describes the knowledge of the complete loci of the purely imaginary characteristic roots in the delay space [15; 32]. These loci are called \"stability switching\" hypersurfaces [33], which are the only possible stability switching locations. Therefore, the CTCR paradigm can be used to compute the purely imaginary characteristic roots and ETDMs by drawing the hypersurfaces. For more details about the CTCR paradigm refer to [15; 32; 33]. **Definition 1**. : _Kernel hypersurfaces: The hypersurfaces contain all the delay points\((\tau_{1},\tau_{2})\), where an imaginary root \((s=\iota\cdot\omega,\,t^{2}=-1)\) is created and the constraint \(0<\tau_{k}\omega<2\pi,\,\,k=1,2\) is satisfied. The points on the kernel hypersurfaces contain the smallest delay compositions, which correspond to all possible imaginary roots._  
**Definition 2**.