Computation　offloading　is　generally　regarded　as　a　promising　technology　to　address　the　problem　of　insufficient　computing　power　in　mobile　devices,　while　simultaneously　satisfying　low-latency　requirements　and　furnishing　exceptional　support　for　intelligent　applications.　However,　with　the　advancement　of　computing-intensive　and　delay-sensitive　application　service　demands,　how　to　assign　applications　with　diversity　requirements　among　various　edge　servers　still　remains　a　challenge.　In　this　paper,　we　propose　a　graph　neural　network　(GNN)-based　dependency-aware　task　scheduling　and　offloading　(GNN-TSO)　scheme　for　MEC-assisted　network　to　effectively　coordinate　wireless　and　computing　resources　for　multiple　applications.　We　first　model　the　dependencies　among　all　tasks　of　an　application　as　a　directed　acyclic　graph　(DAG)　and　formulate　the　dependency-aware　task　scheduling　and　offloading　problem　as　a　combinatorial　optimization　problem　which　is　hard　to　be　solved.　To　capture　the　scalable　features　of　DAG-type　applications,　we　introduce　the　GNN　to　process　tasks　information　of　applications.　Then　we　construct　a　Markov　decision　process　for　the　fine-grained　task　scheduling　and　offloading　strategy　and　apply　policy　gradient　algorithm　to　jointly　optimize　the　task　scheduling　priority　and　computation　offloading　decision.　Simulation　results　show　that　the　proposed　scheme　can　effectively　reduce　the　network　cost　compared　with　other　reference　schemes.　　©　2023　IEEE.