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Routers are one of the major components in supercomputer interconnection networks,which has great influence on the performance and energy consumption in the supercomputer interconnection networks.Given a network topology and routing rule,different resource allocation strategies cause the different numbers of idle routers.A key theoretical question we are concerned about is what the maximum number of idle routers is,i.e.,what the supremum is.For all the possible job scales on a 4 × 4 2d-mesh network,we find the minimum number of used routers.We also enumerate the minimum number of used routers on 5 × 5,6 × 6…10 × 10 2d-mesh networks.Based on the above enumeration results,we propose and prove the theorem of supremum of idle routers on 2d-mesh with dimension-order routing.In order to achieve as high performance as possible or achieve as much energy savings as possible in the supercomputer interconnection networks,sometimes we need to design the corresponding algorithms to find as many idle routers as possible.The supremum of the number of idle routers provides us an evaluation criterion for this kind of heuristic algorithms.