  Recent attention has turned to exploring quantum information within larger
Hilbert spaces by utilizing qudits, which offer increased information capacity
and potential for robust quantum communications. While the efficient generation
of multipartite qudit entanglement is crucial for studying quantum correlations
in high-dimensional Hilbert spaces, the increased dimension makes the circuit
design challanging, especially when the entanglement is generated by heralding
detections. In this work, we demonstrate that the graph picture of linear
quantum networks (LQG picture) can provide a simplified method to generate
qudit multipartite heralded entanglement of high symmetries. The LQG picture
enables the reduction of circuit complexity by directly imposing the state
symmetry onto the circuit structure. Leveraging this insight, we propose
heralded schemes for generating $N$-partite $N$-level anti-symmetric (singlet)
and symmetric (Dicke) states. Our study shed light on the optimal circuit
design of high-dimensional entanglement with a systematic graphical strategy.
