Ensuring edge data integrity is crucial for reliable and secure low-latency services in mobile edge computing. However, existing edge data integrity (EDI) schemes suffer from certain limitations. These include the generation of excessive communication overhead and significant time consumption due to multi-round communications between application vendors and individual edge servers during the integrity verification process. Additionally, there is a notable straggler effect when employing batch verification techniques. To overcome these challenges, this paper presents a pioneering solution scheme for per-edge one-round EDI, referred to as OR-EDI. The proposed scheme effectively mitigates communication overhead, reduces time consumption, and eliminates the straggler effect, while simultaneously maintaining a high level of corruption detection accuracy. To implement OR-EDI, we introduce a novel verification structure known as the Minimal Verification Tree (MVT) and devise algorithms for acquiring Auxiliary Verification Information (AVI) and constructing the MVT. To comprehensively evaluate the performance of OR-EDI, we conduct a rigorous analysis that encompasses its correctness, efficiency, and security aspects. Furthermore, we implement the proposed scheme and perform a comparative assessment against existing EDI schemes, presenting extensive experimental results. These results affirm the effectiveness and efficiency of OR-EDI in effectively addressing the challenges associated with EDI.