Summary: Finding the maximum eigenvalue of a symmetric tensor is an important topic in tensor computation and numerical multilinear algebra. In this paper, we introduce a new class of structured tensors called \(W\)-tensors, which not only extends the well-studied nonnegative tensors by allowing negative entries but also covers several important tensors arising naturally from spectral hypergraph theory. We then show that finding the maximum \(H\)-eigenvalue of an even-order symmetric \(W\)-tensor is equivalent to solving a structured semidefinite program and hence can be validated in polynomial time. This yields a highly efficient semidefinite program algorithm for computing the maximum \(H\)-eigenvalue of \(W\)-tensors and is based on a new structured sums-of-squares decomposition result for a nonnegative polynomial induced by \(W\)-tensors. Numerical experiments illustrate that the proposed algorithm can successfully find the maximum \(H\)-eigenvalue of \(W\)-tensors with dimension up to 10,000, subject to machine precision. As applications, we provide a polynomial time algorithm for computing the maximum \(H\)-eigenvalues of large-size Laplacian tensors of hyperstars and hypertrees, where the algorithm can be up to 13 times faster than the state-of-the-art numerical method introduced by M. Ng, L. Qi, and G. Zhou in 2009 [SIAM J. Matrix Anal. Appl. 31(2009), No. 3, 1090–1099 (2010; Zbl 1197.65036)]. Finally, we also show that the proposed algorithm can be used to test the copositivity of a multivariate form associated with symmetric extended \(Z\)-tensors, whose order may be even or odd.