# Networks

- [Introduction](introduction.md)

- [Construction of Networks](network-constr.md)

  - [`Network<n | edges >: RngIntElt, List → GrphNet, GrphVertSet, GrphEdgeSet`](network-constr.md#constructor-constructor-network-rngintelt-list-grphnet-grphvertset-grphedgeset)

  - [`Network<S | edges >: SetEnum, List → GrphNet, GrphVertSet, GrphEdgeSet`](network-constr.md#constructor-constructor-network-setenum-list-grphnet-grphvertset-grphedgeset)

  - [`Network<S | edges >: SetIndx, List → GrphNet, GrphVertSet, GrphEdgeSet`](network-constr.md#constructor-constructor-network-setindx-list-grphnet-grphvertset-grphedgeset)

  - [`Example: GrphNet Constr`](network-constr.md#example-ex-4c4672)

  - [Magma Output: Printing of a Network](network-constr.md#magma-output-printing-of-a-network)

    - [`Example: GrphNet Constr2`](network-constr.md#example-ex-5be897)

- [Standard Construction for Networks](network-stand-constr.md)

  - [Subgraphs](network-stand-constr.md#subgraphs)

    - [`sub< N | list >: GrphNet, List → GrphNet, GrphVertSet, GrphEdgeSet`](network-stand-constr.md#constructor-constructor-sub-grphnet-list-grphnet-grphvertset-grphedgeset)

    - [`Example: Constr Sub Network`](network-stand-constr.md#example-ex-48c2fa)

  - [Incremental Construction: Adding Edges](network-stand-constr.md#incremental-construction-adding-edges)

    - [`N + < [ u, v ], c >: GrphNet, < [ GrphVert, GrphVert ], RngIntElt > → GrphNet, GrphEdge`](network-stand-constr.md#operation-op-plus-grphnet-grphvert-grphvert-rngintelt)

    - [`N + { < [ u, v ], c > }: GrphNet, {  < [ GrphVert, GrphVert ], RngIntElt > } → GrphNet`](network-stand-constr.md#operation-op-plus-grphnet-grphvert-grphvert-rngintelt-2)

    - [`N + [ < [ u, v ], c > ]: GrphNet, [ < [ GrphVert, GrphVert ], RngIntElt > ] → GrphNet`](network-stand-constr.md#operation-op-plus-grphnet-grphvert-grphvert-rngintelt-3)

    - [`N +:= < [ u, v ], c >: GrphNet, < [ GrphVert, GrphVert ], RngIntElt >`](network-stand-constr.md#operation-operation-grphnet-grphvert-grphvert-rngintelt)

    - [`N +:= { < [ u, v ], c > }: GrphNet, {  < [ GrphVert, GrphVert ], RngIntElt > }`](network-stand-constr.md#operation-operation-grphnet-grphvert-grphvert-rngintelt-2)

    - [`N +:= [ < [ u, v ], c > ]: GrphNet, [ < [ GrphVert, GrphVert ], RngIntElt > ]`](network-stand-constr.md#operation-operation-grphnet-grphvert-grphvert-rngintelt-3)

    - [`AddEdge(N, u, v, c): GrphNet, GrphVert, GrphVert, RngIntElt → GrphNet, GrphEdge`](network-stand-constr.md#function-multigraph-addedgecap)

    - [`AddEdge(N, u, v, c, l): GrphNet, GrphVert, GrphVert, RngIntElt, . → GrphNet, GrphEdge`](network-stand-constr.md#function-multigraph-addedgecaplabel)

    - [`AddEdges(N, S): GrphNet, {  < [  GrphVert, GrphVert ], RngIntElt > } → GrphNet`](network-stand-constr.md#function-addedges-grphnet-grphvert-grphvert-rngintelt)

    - [`AddEdges(N, S): GrphNet, [ < [  GrphVert, GrphVert ], RngIntElt > ] → GrphNet`](network-stand-constr.md#function-addedges-grphnet-grphvert-grphvert-rngintelt-2)

    - [`AddEdge(~N, u, v, c): GrphNet, GrphVert, GrphVert, RngIntElt`](network-stand-constr.md#function-addedge-grphnet-grphvert-grphvert-rngintelt-ref)

    - [`AddEdge(~N, u, v, c, l): GrphNet, GrphVert, GrphVert, RngIntElt, .`](network-stand-constr.md#function-addedge-grphnet-grphvert-grphvert-rngintelt-ref-2)

    - [`AddEdges(~N, S): GrphNet, {  < [  GrphVert, GrphVert ], RngIntElt > }`](network-stand-constr.md#function-addedges-grphnet-grphvert-grphvert-rngintelt-ref)

    - [`AddEdges(~N, S): GrphNet, [ < [  GrphVert, GrphVert ], RngIntElt > ]`](network-stand-constr.md#function-addedges-grphnet-grphvert-grphvert-rngintelt-ref-2)

  - [Union of Networks](network-stand-constr.md#union-of-networks)

- [Maximum Flow and Minimum Cut](flow.md)

  - [`MinimumCut(s, t : parameters): GrphVert, GrphVert → SeqEnum, RngIntElt`](flow.md#function-minimumcut-grphvert-grphvert)

  - [`MinimumCut(Ss, Ts : parameters): [ GrphVert ], [ GrphVert ] → SeqEnum, RngIntElt`](flow.md#function-minimumcut-grphvert-grphvert-2)

  - [`MaximumFlow(s, t : parameters): GrphVert, GrphVert → RngIntElt, SeqEnum`](flow.md#function-multigraph-maximumflow)

  - [`MaximumFlow(Ss, Ts : parameters): [ GrphVert ], [ GrphVert ] → RngIntElt, SeqEnum`](flow.md#function-maximumflow-grphvert-grphvert)

  - [`Flow(e): GrphEdge → RngIntElt`](flow.md#function-flow-grphedge)

  - [`Flow(u, v): GrphVert, GrphVert → RngIntElt`](flow.md#function-flow-grphvert-grphvert)

  - [`Example: Flow`](flow.md#example-ex-00952b)