## 1 Introduction  
Vehicular platoon technology is a promising approach to improving traffic capacity, enhancing highway safety, and reducing exhaust emissions and fuel consumption [1; 2]. A vehicular platoon consists of a group of coordinated vehicles that are connected autonomously by using vehicle-to-vehicle (V2V) and vehicle-to-infrastructure (V2I) communication technologies [3]. To maintain a stable vehicular platoon, it is crucial to ensure both the internal stability and string stability [4]. The internal stability refers to the convergence of the trajectories of all the vehicles within the platoon [5]. On the other hand, the string stability implies that disturbances should not amplify when propagating along the vehicle string [6]. In the vehicular platoon, time delays are introduced due to the sensing time and limited communication bandwidth, which are unavoidable aspects of using the sensors and wireless communication [7; 8]. These time delays are widely recognized as the primary factors that can significantly degrade the performance of the vehicular platoon and potentially lead to the internal instability or string instability [9]. Hence, it is essential to study the stability of the vehicular platoon with sensing delay and communication delay. Numerous studies have been conducted on delay-dependent stability analysis of vehicular platoons. For investigating the internal stability of the system with multiple delays, both the exponential convergence rates of the system's states and delay margin are estimated by using the Lyapunov-Razumikhin theorem [10]. For the vehicular platoon with heterogeneous communication delays, the sufficient internal stability conditions are solvedby using the Lyapunov-Krasovskii method [11]. The Nyquist criterion has been utilized to investigate the internal stability affected by the communication delay and determine the maximum allowable time delay [12]. Most existing studies of analyzing the internal stability of the vehicular platoon with time delays rely on the Lyapunov-Razumikhin and Lyapunov-Krasovskii functions [13; 14]. However, these studies can only obtain the sufficient stability conditions, rather than the necessary and sufficient conditions, resulting in relatively conservative results. Thus, the exact (non-conservative) time delay margins (ETDMs) could barely be solved using the above-mentioned methods. Additionally, since these studies involve solving the linear matrix inequalities, the computational burden exponentially increases with the dimension of the matrix. For solving the ETDMs of the vehicular platoon, a crucial stability paradigm called cluster treatment of characteristic root (CTCR) is deployed in a different way [15]. The CTCR paradigm provides a comprehensive understanding of the loci of the purely imaginary characteristic roots of the closed-loop platoon dynamics and offers a systematic method of determining the exact and exhaustive stability for both the single-delay and multi-delay cases. These loci represent the potential delay margins in the delay space, and they are located on the kernel and offspring hypersurfaces within the CTCR paradigm. The CTCR paradigm enables a graphical representation of the delay margins and can be implemented numerically [16]. By combining the CTCR paradigm with the Rekasius transformation, a necessary and sufficient internal stability condition is derived for the vehicular platoon with time delays [17]. For the vehicular platoon under generic communication topologies with two time delays, a resultant matrix-based approach is proposed to determine the kernel and offspring hypersurfaces of the CTCR paradigm [18; 19]. Though the CTCR paradigm can be used to derive the ETDMs of the vehicular platoon, it bears a heavy computational burden due to its numerical and time-consuming frequency sweeping process [20]. The frequency sweeping process involves high-dimensional matrix computation, and as the dimensions of the matrices in the CTCR paradigm increase, the computational burden grows exponentially.