To reveal the inherent relationship between the closed-loop stability of delayed vehicle platoons and communication topologies, a universal method of searching the most exigent eigenvalue (MEE) is proposed for second-order vehicle platoons under arbitrary communication topologies. First, for the case where all eigenvalues of the Laplacian matrix are real, by analyzing the monotonicity relationship between the maximum allowable delay and the eigenvalues, it is proven that the MEE of the second-order vehicle platoon must be the maximum eigenvalue of the Laplacian matrix. Then, for the case where some eigenvalues are complex conjugates, it is found that the maximum allowable delays corresponding to a pair of complex conjugate eigenvalues are equal in size, and an analytical expression for the maximum allowable delay is provided. Furthermore, during the analysis of the aforementioned monotonic relationship between the maximum allowable delay and the eigenvalues, the influence of the magnitude and phase of the complex conjugate eigenvalues on the MEE is revealed, and a concise MEE search rule is proposed. The results show that the proposed theoretical method is validated through simulation examples. Compared with traditional traversal methods of calculating the maximum allowable delay, the proposed method significantly reduces the computational burden. This method of searching the MEE is applicable to the closed-loop stability analysis of second-order vehicle platoons under arbitrary communication topologies and demonstrates universality.