# MultifunctionGraph SΒΆ

- S: SetCategory

allows us to model graph theory

- *: (%, %) -> MultifunctionGraph Product(S, S)
- tensor product : the tensor product
`G*H`

of graphs`G`

and`H`

is a graph such that the vertex set of`G*H`

is the Cartesian product`V`

(`G`

) times`V`

(`H`

); and any two vertices (`u`

, u’) and (`v`

,`v'`

) are adjacent in`G`

times`H`

if and only if u’ is adjacent with`v'`

and`u`

is adjacent with`v`

. - +: (%, %) -> %
- from FiniteGraph S
- =: (%, %) -> Boolean
- from BasicType

- ~: % -> %
- The complement or inverse of a graph is a graph on the same vertices such that there is an arrow if and only if there is not an arrow in its compliment. That is, it is the compliment of the arrows but is not the set complement. for more information see: http://en.wikipedia.org/wiki/Complement_graph
- ~=: (%, %) -> Boolean
- from BasicType
- addArrow!: (%, Record(name: String, arrType: NonNegativeInteger, fromOb: NonNegativeInteger, toOb: NonNegativeInteger, xOffset: Integer, yOffset: Integer, map: List NonNegativeInteger)) -> %
- from FiniteGraph S
- addArrow!: (%, String, NonNegativeInteger, NonNegativeInteger) -> %
- from FiniteGraph S
- addArrow!: (%, String, NonNegativeInteger, NonNegativeInteger, List NonNegativeInteger) -> %
- from FiniteGraph S
- addArrow!: (%, String, S, S) -> %
- from FiniteGraph S
- addObject!: (%, Record(value: S, posX: NonNegativeInteger, posY: NonNegativeInteger)) -> %
- from FiniteGraph S
- addObject!: (%, S) -> %
- from FiniteGraph S
- adjacencyMatrix: % -> Matrix NonNegativeInteger
- from FiniteGraph S

- apply: (%, NonNegativeInteger, NonNegativeInteger) -> NonNegativeInteger
`apply '`

function’ represented by this graph to vertex index ‘a’- arrowName: (%, NonNegativeInteger, NonNegativeInteger) -> String
- from FiniteGraph S
- arrowsFromArrow: (%, NonNegativeInteger) -> List NonNegativeInteger
- from FiniteGraph S
- arrowsFromNode: (%, NonNegativeInteger) -> List NonNegativeInteger
- from FiniteGraph S
- arrowsToArrow: (%, NonNegativeInteger) -> List NonNegativeInteger
- from FiniteGraph S
- arrowsToNode: (%, NonNegativeInteger) -> List NonNegativeInteger
- from FiniteGraph S

- cartesian: (%, %) -> MultifunctionGraph Product(S, S)
- Cartesian product doubles the size of next list in each object, that is it produces two arrows out of every node

- closedCartesian: (%, %, (S, S) -> S) -> %
- Cartesian product doubles the size of next list in each object, that is it produces two arrows out of every node

- closedTensor: (%, %, (S, S) -> S) -> %
- as tensor product but returns %.

- coAdjoint: (%, List NonNegativeInteger) -> Union(List NonNegativeInteger, failed)
- given a mapping from this graph this function tries to calculate a unique reverse mapping back to this graph
- coerce: % -> OutputForm
- from CoercibleTo OutputForm

- coerce: PermutationGroup S -> %
`coerce PermutationGroup`

to graph which represents the generators of the group

- contraAdjoint: (%, List NonNegativeInteger) -> Union(List NonNegativeInteger, failed)
- given a mapping from this graph this function tries to calculate a unique reverse mapping back to this graph
- createWidth: NonNegativeInteger -> NonNegativeInteger
- from FiniteGraph S
- createX: (NonNegativeInteger, NonNegativeInteger) -> NonNegativeInteger
- from FiniteGraph S
- createY: (NonNegativeInteger, NonNegativeInteger) -> NonNegativeInteger
- from FiniteGraph S
- cycleClosed: (List S, String) -> %
- from FiniteGraph S
- cycleOpen: (List S, String) -> %
- from FiniteGraph S
- deepDiagramSvg: (String, %, Boolean) -> Void
- from FiniteGraph S
- diagramHeight: % -> NonNegativeInteger
- from FiniteGraph S
- diagramsSvg: (String, List %, Boolean) -> Void
- from FiniteGraph S
- diagramSvg: (String, %, Boolean) -> Void
- from FiniteGraph S
- diagramWidth: % -> NonNegativeInteger
- from FiniteGraph S
- distance: (%, NonNegativeInteger, NonNegativeInteger) -> Integer
- from FiniteGraph S
- distanceMatrix: % -> Matrix Integer
- from FiniteGraph S
- flatten: DirectedGraph % -> %
- from FiniteGraph S
- getArrowIndex: (%, NonNegativeInteger, NonNegativeInteger) -> NonNegativeInteger
- from FiniteGraph S
- getArrows: % -> List Record(name: String, arrType: NonNegativeInteger, fromOb: NonNegativeInteger, toOb: NonNegativeInteger, xOffset: Integer, yOffset: Integer, map: List NonNegativeInteger)
- from FiniteGraph S
- getVertexIndex: (%, S) -> NonNegativeInteger
- from FiniteGraph S
- getVertices: % -> List Record(value: S, posX: NonNegativeInteger, posY: NonNegativeInteger)
- from FiniteGraph S
- hash: % -> SingleInteger
- from SetCategory
- hashUpdate!: (HashState, %) -> HashState
- from SetCategory
- incidenceMatrix: % -> Matrix Integer
- from FiniteGraph S
- inDegree: (%, NonNegativeInteger) -> NonNegativeInteger
- from FiniteGraph S
- initial: () -> %
- from FiniteGraph S
- isAcyclic?: % -> Boolean
- from FiniteGraph S
- isDirected?: () -> Boolean
- from FiniteGraph S
- isDirectSuccessor?: (%, NonNegativeInteger, NonNegativeInteger) -> Boolean
- from FiniteGraph S
- isFixPoint?: (%, NonNegativeInteger) -> Boolean
- from FiniteGraph S
- isFunctional?: % -> Boolean
- from FiniteGraph S
- isGreaterThan?: (%, NonNegativeInteger, NonNegativeInteger) -> Boolean
- from FiniteGraph S
- kgraph: (List S, String) -> %
- from FiniteGraph S
- laplacianMatrix: % -> Matrix Integer
- from FiniteGraph S
- latex: % -> String
- from SetCategory

- limit: (%, NonNegativeInteger, NonNegativeInteger) -> Loop
- apply ‘function’ represented by this graph to ‘a’ repeatedly until we reach a loop which is returned as a sequence of vertex indexes.
- loopsArrows: % -> List Loop
- from FiniteGraph S
- loopsAtNode: (%, NonNegativeInteger) -> List Loop
- from FiniteGraph S
- loopsNodes: % -> List Loop
- from FiniteGraph S
- looseEquals: (%, %) -> Boolean
- from FiniteGraph S
- map: (%, List NonNegativeInteger, List S, Integer, Integer) -> %
- from FiniteGraph S
- mapContra: (%, List NonNegativeInteger, List S, Integer, Integer) -> %
- from FiniteGraph S
- max: % -> NonNegativeInteger
- from FiniteGraph S
- max: (%, List NonNegativeInteger) -> NonNegativeInteger
- from FiniteGraph S
- merge: (%, %) -> %
- from FiniteGraph S
- min: % -> NonNegativeInteger
- from FiniteGraph S
- min: (%, List NonNegativeInteger) -> NonNegativeInteger
- from FiniteGraph S

- multifunctionGraph: (List Record(value: S, posX: NonNegativeInteger, posY: NonNegativeInteger), List Record(name: String, arrType: NonNegativeInteger, fromOb: NonNegativeInteger, toOb: NonNegativeInteger, xOffset: Integer, yOffset: Integer, map: List NonNegativeInteger)) -> %
- constructor for graph with given objects and arrows more objects and arrows can be added later if required.

- multifunctionGraph: (List S, List List NonNegativeInteger) -> %
- constructor for graph with given objects and adjacency matrix.

- multifunctionGraph: List Permutation S -> %
- construct graph from a list of permutations.

- multifunctionGraph: List Record(value: S, posX: NonNegativeInteger, posY: NonNegativeInteger, next: List NonNegativeInteger, map: List List NonNegativeInteger) -> %
- constructor for graph with given objects more objects and arrows can be added later if required.

- multifunctionGraph: List S -> %
- constructor for graph with given list of object names. Use this version of the constructor if you don
`'t`

intend to create diagrams and therefore don`'t`

care about`x`

,`y`

coordinates. more objects and arrows can be added later if required. - nodeFromArrow: (%, NonNegativeInteger) -> List NonNegativeInteger
- from FiniteGraph S
- nodeFromNode: (%, NonNegativeInteger) -> List NonNegativeInteger
- from FiniteGraph S
- nodeToArrow: (%, NonNegativeInteger) -> List NonNegativeInteger
- from FiniteGraph S
- nodeToNode: (%, NonNegativeInteger) -> List NonNegativeInteger
- from FiniteGraph S
- outDegree: (%, NonNegativeInteger) -> NonNegativeInteger
- from FiniteGraph S
- routeArrows: (%, NonNegativeInteger, NonNegativeInteger) -> List NonNegativeInteger
- from FiniteGraph S
- routeNodes: (%, NonNegativeInteger, NonNegativeInteger) -> List NonNegativeInteger
- from FiniteGraph S
- spanningForestArrow: % -> List Tree Integer
- from FiniteGraph S
- spanningForestNode: % -> List Tree Integer
- from FiniteGraph S
- spanningTreeArrow: (%, NonNegativeInteger) -> Tree Integer
- from FiniteGraph S
- spanningTreeNode: (%, NonNegativeInteger) -> Tree Integer
- from FiniteGraph S
- subdiagramSvg: (Scene SCartesian 2, %, Boolean, Boolean) -> Void
- from FiniteGraph S
- terminal: S -> %
- from FiniteGraph S

- toCayleyGraph: (List Permutation S, Boolean) -> MultifunctionGraph String
- convert permutation generators to a Cayley graph permList should contain generator permutations and should not contain identity permutation. if permutationNames then names generated represent permutation

- toCayleyGraph: PermutationGroup S -> MultifunctionGraph String
- convert PermutationGroup to a Cayley graph

- toPermutation: % -> PermutationGroup NonNegativeInteger
- generates a permutation group from this graph assumes this graph represents a valid group
- unit: (List S, String) -> %
- from FiniteGraph S