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# Graph

## Functions and methods returning a graph:

- `addEdge(Digraph,Set)` -- see addEdge -- A method for adding edges to a graph
- addEdges'(Graph,List) (missing documentation)
- `addVertices(Digraph,List)` -- see addVertex -- A method for adding a set of vertices to a graph
- addVertices(Graph,List) (missing documentation)
- `barbellGraph(ZZ)` -- see barbellGraph -- Returns the barbell graph
- `barycenter(Graph)` -- see barycenter -- Returns the barycenter of a grah
- `cartesianProduct(Graph,Graph)` -- see cartesianProduct -- Computes the cartesian product of two graphs
- `circularLadder(ZZ)` -- see circularLadder -- Returns a circular ladder graph
- `cocktailParty(ZZ)` -- see cocktailParty -- Returns a cocktail party graph
- `complementGraph(Graph)` -- see complementGraph -- Returns the complement of a graph
- `completeGraph(ZZ)` -- see completeGraph -- Constructs a complete graph
- `completeMultipartiteGraph(List)` -- see completeMultipartiteGraph -- constructs a complete multipartite graph
- `crownGraph(ZZ)` -- see crownGraph -- Returns a crown graph
- `cycleGraph(ZZ)` -- see cycleGraph -- Constructs a cycle graph
- `deleteEdges(Graph,List)` -- see deleteEdges -- Deletes a list of edges from a graph
- deleteEdges(Digraph,List) (missing documentation)
- `deleteVertex(Graph,Thing)` -- see deleteVertex -- a method for deleting the vertex of a graph
- `directProduct(Graph,Graph)` -- see directProduct -- Computes the direct product of two graphs
- `disjointUnion(List)` -- see disjointUnion -- Returns the disjoint union of a list of graphs.
- `doubleStar(ZZ,ZZ)` -- see doubleStar -- returns a double star graph
- `friendshipGraph(ZZ)` -- see friendshipGraph -- Returns a friendship Graph
- `generalizedPetersenGraph(ZZ,ZZ)` -- see generalizedPetersenGraph -- Returns a generalized petersen graph
- `graph(HashTable)` -- see graph -- Constructs a simple graph
- `graph(List)` -- see graph -- Constructs a simple graph
- `graph(List,List)` -- see graph -- Constructs a simple graph
- `graph(List,Matrix)` -- see graph -- Constructs a simple graph
- `graph(Matrix)` -- see graph -- Constructs a simple graph
- `graphComposition(Graph,Graph)` -- see graphComposition -- A method for composing two graphs
- `graphLibrary(String)` -- see graphLibrary -- constructs a graph of a type specified in the string input
- `graphPower(Graph,ZZ)` -- see graphPower -- constructs a graph raised to a power
- `indexLabelGraph(Graph)` -- see indexLabelGraph -- Relabels the vertices of a graph or digraph according to their indices, indexed from 0.
- `inducedSubgraph(Graph,List)` -- see inducedSubgraph -- A method for finding the induced subgraph of any Graph or Digraph
- `kneserGraph(ZZ,ZZ)` -- see kneserGraph -- constructs a kneser graph of specified size
- `ladderGraph(ZZ)` -- see ladderGraph -- Returns a ladder graph
- `lineGraph(Graph)` -- see lineGraph -- Returns the line graph of an undirected graph
- `lollipopGraph(ZZ,ZZ)` -- see lollipopGraph -- constructs a lollipop graph
- `monomialGraph(MonomialIdeal,ZZ)` -- see monomialGraph -- Returns a monomial graph
- `pathGraph(ZZ)` -- see pathGraph -- A method that makes a path graph
- `rattleGraph(ZZ,ZZ)` -- see rattleGraph -- Returns a rattle graph
- `reindexBy(Graph,String)` -- see reindexBy -- reindexes the vertices according to the input ordering.
- `spanningForest(Graph)` -- see spanningForest -- constructs a spanning forest of a graph
- `starGraph(ZZ)` -- see starGraph -- Returns a star graph
- `strongProduct(Graph,Graph)` -- see strongProduct -- a method for taking the strong product of two graphs
- `thresholdGraph(List)` -- see thresholdGraph -- A method that generates a threshold graph from a binary list
- `underlyingGraph(Digraph)` -- see underlyingGraph -- Returns the underlying graph of a digraph
- `vertexMultiplication(Graph,Thing,Thing)` -- see vertexMultiplication
- `wheelGraph(ZZ)` -- see wheelGraph -- Constructs a wheel graph
- `windmillGraph(ZZ,ZZ)` -- see windmillGraph -- Constructs a windmill graph

## Methods that use a graph:

- `bipartiteColoring(Graph)` -- see bipartiteColoring -- Returns a coloring of a bipartite graph
- `center(Graph)` -- see center -- Returns the center of a graph
- `chromaticNumber(Graph)` -- see chromaticNumber -- Computes the chromatic number of a graph
- `cliqueComplex(Graph)` -- see cliqueComplex -- Returns the clique complex of a graph
- `cliqueNumber(Graph)` -- see cliqueNumber -- Returns the clique number of a graph
- `closedNeighborhood(Graph,Thing)` -- see closedNeighborhood -- Returns the closed neighborhood of a vertex of a graph
- `clusteringCoefficient(Graph)` -- see clusteringCoefficient -- a method for computing the clustering coefficient of a Graph
- `clusteringCoefficient(Graph,Thing)` -- see clusteringCoefficient -- a method for computing the clustering coefficient of a Graph
- connectedComponents(Graph) -- Computes the connected components of a graph
- `coverIdeal(Graph)` -- see coverIdeal -- Returns the vertex cover ideal of a graph
- `criticalEdges(Graph)` -- see criticalEdges -- Finds the critical edges of a graph
- `degeneracy(Graph)` -- see degeneracy -- Computes the degeneracy of a graph
- degree(Graph,Thing) (missing documentation)
- `degreeCentrality(Graph,Thing)` -- see degreeCentrality -- Returns the degreeCentrality of a vertex of a graph
- `degreeSequence(Graph)` -- see degreeSequence -- the degree sequence of a graph
- `density(Graph)` -- see density -- computes the density of a graph
- diameter(Graph) -- Computes the diameter of a graph
- `eccentricity(Graph,Thing)` -- see eccentricity -- Returns the eccentricity of a vertex of a graph
- `edgeConnectivity(Graph)` -- see edgeConnectivity -- computes the edge connectivity of a graph
- `edgeCuts(Graph)` -- see edgeCuts -- returns the edge cuts of a graph
- `edgeIdeal(Graph)` -- see edgeIdeal -- returns the edge ideal of a graph
- `edges(Graph)` -- see edges -- Returns the edges of a digraph or graph
- `expansion(Graph)` -- see expansion -- returns the expansion of a graph
- `girth(Graph)` -- see girth -- A method for computing the girth of a graph
- `hasEulerianTrail(Graph)` -- see hasEulerianTrail -- determines whether a graph or a digraph has an Eulerian trail
- `hasOddHole(Graph)` -- see hasOddHole -- checks whether a graph has a odd hole
- `incidenceMatrix(Graph)` -- see incidenceMatrix -- computes the incidence matrix of a graph
- `independenceComplex(Graph)` -- see independenceComplex -- constructs the independence complex of a graph
- `independenceNumber(Graph)` -- see independenceNumber -- computes the independence number of a graph
- `isBipartite(Graph)` -- see isBipartite -- determines whether a graph is bipartite
- `isChordal(Graph)` -- see isChordal -- checks whether a graph is chordal
- `isCM(Graph)` -- see isCM -- determines if a graph is Cohen-Macaulay
- `isConnected(Graph)` -- see isConnected -- determines whether a graph is connected
- `isCyclic(Graph)` -- see isCyclic -- determines whether a graph is cyclic
- `isEulerian(Graph)` -- see isEulerian -- determines if a graph or digraph is Eulerian
- `isForest(Graph)` -- see isForest -- determines whether a graph is a forest
- `isLeaf(Graph,Thing)` -- see isLeaf -- determines whether a vertex is a leaf
- `isPerfect(Graph)` -- see isPerfect -- checks whether a graph is perfect
- `isRegular(Graph)` -- see isRegular -- determines whether a graph is regular
- `isRigid(Graph)` -- see isRigid -- checks if a graph is rigid
- `isSimple(Graph)` -- see isSimple -- checks if a graph is simple
- `isTree(Graph)` -- see isTree -- determines whether a graph is a tree
- `laplacianMatrix(Graph)` -- see laplacianMatrix -- Returns the laplacian matrix of a graph
- `leaves(Graph)` -- see leaves -- lists the leaves of a tree graph
- `minimalDegree(Graph)` -- see minimalDegree -- computes the minimal degree of a graph
- `minimalVertexCuts(Graph)` -- see minimalVertexCuts -- finds the minimal vertex cuts of a graph
- `neighbors(Graph,Thing)` -- see neighbors -- returns the neighbors of a vertex in a graph
- `nonneighbors(Graph,Thing)` -- see nonneighbors -- returns the non-neighbors of a vertex in a graph
- `numberOfComponents(Graph)` -- see numberOfComponents -- computes the number of connected components of a graph
- `numberOfTriangles(Graph)` -- see numberOfTriangles -- counts how many subtriangles are present in a graph
- `radius(Graph)` -- see radius -- Returns the radius of a graph
- `spectrum(Graph)` -- see spectrum -- Returns the spectrum of a graph
- `vertexConnectivity(Graph)` -- see vertexConnectivity -- computes the vertex connectivity of a graph
- `vertexCoverNumber(Graph)` -- see vertexCoverNumber -- returns the vertex cover number of a graph
- `vertexCovers(Graph)` -- see vertexCovers -- returns a list of the minimal vertex covers of a graph
- `vertexCuts(Graph)` -- see vertexCuts -- lists all the vertex cuts of a graph
- `writeDotFile(String,Graph)` -- see writeDotFile -- Writes a graph to a dot file with a specified filename

## For the programmer

The object Graph is a type, with ancestor classes Digraph < HashTable < Thing.

The source of this document is in /build/macaulay2-88fgJW/macaulay2-1.25.11+ds/M2/Macaulay2/packages/Graphs.m2:1775:0.