<p>This paper investigates the bipartite synchronization problem of discrete-time coupled harmonic oscillators over co-opetitive networks with and without time delay. Two distributed control protocols are developed to achieve bipartite synchronization, where one is based on relative-velocity feedback without time delay and the other is based on relative-position feedback in the presence of time delay. By integrating algebraic graph theory, matrix analysis (including the Gershgorin circle theorem), and frequency-domain stability criteria such as the generalized Nyquist criterion, sufficient conditions are established for bipartite synchronization in coupled harmonic oscillators under directed and undirected network topologies. The study further considers the effect of time delay and shows that the maximum admissible sampling period for bipartite synchronization decreases as the time delay increases. Finally, numerical simulations are provided to validate the theoretical results.</p>