With the rapid growth of data volume, the escalating complexity of data businesses, and the increasing reliance on the Internet for daily life and production, the scale of data centers is constantly expanding. The data center network (DCN) is a bridge connecting large-scale servers in data centers for large-scale distributed computing. How to build a DCN structure that is flexible and cost-effective, while maintaining its topological properties unchanged during network expansion has become a challenging issue. In this paper, we propose an expandable and cost-effective DCN, namely HHCube, which is based on the half hypercube structure. Further, we analyze some characteristics of HHCube, including connectivity, diameter, and bisection bandwidth of the HHCube. We also design an efficient algorithm to find the shortest path between any two distinct nodes and present a fault-tolerant routing scheme to obtain a fault-tolerant path between any two distinct fault-free nodes in HHCube. Meanwhile, we present two local diagnosis algorithms to determine the status of nodes in HHCube under the PMC model and MM* model, respectively. Our results demonstrate that despite the presence of up to 25% faulty nodes in HHCube, both algorithms achieve a correct diagnosis rate exceeding 90%. Finally, we compare HHCube with state-of-the-art DCNs including Fat-Tree, DCell, BCube, Ficonn, and HSDC, and the experimental results indicate that the HHCube is an excellent candidate for constructing expandable and cost-effective DCNs.