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The valley-Chern and spin-valley-Chern numbers are the key concepts invalleytronics. They are topological numbers in the Dirac theory but not in thetight-binding model. We analyze the bulk-edge correspondence between the twophases which have the same Chern and spin-Chern numbers but differentvalley-Chern and spin-valley-Chern numbers. The edge state between them istopologically trivial in the tight-binding model but is shown to be as robustas the topological edge. We construct Y-junctions made of topological edges.They satisfy the topological Kirchhoff law, where the topological charges areconserved at the junction. We may interpret a Y-junction as a scatteringprocess of particles which have four topological numbers. It would be amilestone of future topological electronics.

Topological Kirchhoff Law and Bulk-Edge Correspondence for Valley-Chern and Spin-Valley-Chern Numbers