Graphs are first-class citizens in the Wolfram Language and can be used as input, output, in programs, and in documents. Undirected and directed graphs are treated uniformly and support a number of standard properties for vertices and edges. Importantly, graphs also support custom properties for modeling or computational flexibility. Graphs can be converted to a number of different representations, including matrices. Graphs can be exported with high fidelity to numerous file formats.  Graphs can be constructed in a variety of ways. They can be built from vertices and edges directly in a symbolic form. They can come from built-in curated collections of theoretical or empirical graphs. Special graphs can be generated from parametric specifications. Random graphs following a variety of graph distributions allow you to build simulated internets or citation graphs and test algorithms. Graphs can be fully specified by several types of matrices, or they can be imported from numerous supported file formats. Graphs can also be constructed in several steps by performing operations on graphs.

Graph — represent a general graph, or create it from vertices and edges

UndirectedEdge — an undirected edge () (also entered as <->)

DirectedEdge — a directed edge () (also entered as ->)

Basic Properties

VertexList, EdgeList — the list of vertices and edges in the graph

VertexIndex, EdgeIndex — the index for a vertex or edge in a graph

AdjacencyList  ▪  IncidenceList  ▪  EdgeRules  ▪  PlanarFaceList

Matrix Representations »

AdjacencyMatrix, AdjacencyGraph — matrix representation and graph construction

IncidenceMatrix  ▪  KirchhoffMatrix  ▪  WeightedAdjacencyMatrix  ▪  ...

Graph Annotations »

AnnotationValue — get and set vertex or edge annotation values

EdgeWeight  ▪  VertexWeight  ▪  EdgeCapacity  ▪  VertexCapacity  ▪  ...

Import and Export Graphs »

Import, Export — import and export graphs

"GraphML"  ▪  "GXL"  ▪  "Graphlet"  ▪  "Pajek"  ▪  "TGF"  ▪  "DOT"  ▪  "DIMACS"  ▪  "Graph6"  ▪  "Sparse6"  ▪  "LEDA"

Curated Graphs

GraphData — collection of theoretical graphs

ExampleData — collection of empirical graphs

Polyhedron Chemical Word Country Isotope Species ...

Parametric Graphs

CompleteGraph — generate a complete or a complete k-partite graph

BuckyballGraph  ▪  ButterflyGraph  ▪  CirculantGraph  ▪  CompleteKaryTree  ▪  CycleGraph  ▪  DeBruijnGraph  ▪  GridGraph  ▪  HararyGraph  ▪  HypercubeGraph  ▪  KaryTree  ▪  KnightTourGraph  ▪  PetersenGraph  ▪  StarGraph  ▪  TorusGraph  ▪  TuranGraph  ▪  WheelGraph

Structured Graphs

PathGraph — general directed or undirected path

TreeGraph — general directed or undirected tree

PlanarGraph — general directed or undirected planar graph

Graphs from Data

RelationGraph — generate a graph based on data and a binary relation

NearestNeighborGraph — generate the k-nearest neighbor graph for general elements

NestGraph — generate a nested function graph

ExpressionGraph — generate a graph of the tree structure of expressions

ClusteringTree — generate a tree from hierarchical clustering of elements

CayleyGraph  ▪  MoleculeGraph  ▪  MeshConnectivityGraph  ▪  MorphologicalGraph

Random Graphs »

RandomGraph — generate random graphs following a graph distribution

GraphPropertyDistribution  ▪  UniformGraphDistribution  ▪  ...

Operations and Modifications of Graphs »

Subgraph — extract subgraphs

NeighborhoodGraph  ▪  GraphUnion  ▪  VertexAdd  ▪  EdgeAdd  ▪  ...