# Wolfram Language & System 11.0 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
WOLFRAM LANGUAGE GUIDE

# Graph Properties & Measurements

Many algorithms and procedures require graphs with certain properties. These can be basic properties, such as being undirected, or deeper topology properties, such as being connected or acyclic. In some areas, a key problem is to decide whether two graphs are the same if the vertex names are replaced, i.e. to test whether they are isomorphic.

The Wolfram Language supports a broad range of measures that characterize graphs, from simple measures, such as the number of vertices and edges, which tell the size and sparsity of a graph, to vertex degrees, which tell how locally well connected each vertex is. Other measures include the geodesic distances in a graph or centrality measures, which give a measure of how central in the overall graph each vertex is; for example, PageRank and HITS are measures used to order web page importance as returned from a search engine.

## ReferenceReference

### Basic Properties

GraphQ test whether an expression is a graph object

DirectedGraphQ, UndirectedGraphQ test whether a graph is directed or undirected

MultigraphQ, MixedGraphQ test whether a graph is a multigraph or a mixed graph

### Structural Properties

SimpleGraphQ test whether a graph is simple

AcyclicGraphQ test whether a graph is acyclic

### Graph Isomorphism

IsomorphicGraphQ test whether two graphs are the same after vertex renaming

FindGraphIsomorphism find the graph isomorphism as a list of rules

### Basic Measures

VertexCount, EdgeCount give the number of vertices and edges in a graph

VertexDegree give the number of edges for each vertex

### Distance Measures

GraphDistance the length of the shortest path between two vertices

### Connectivity Measures

VertexConnectivity the number of vertex-independent paths between two vertices

EdgeConnectivity the number of edge-independent paths between two vertices

GraphDensity fraction of edges to the possible edges in a graph

GraphLinkEfficiency how tightly connected a graph is compared to number of edges

### Centrality Measures

ClosenessCentrality inverse average distance to every other vertex

BetweennessCentrality fraction of shortest paths that pass through the vertex

### Reciprocity and Transitivity

GraphReciprocity fraction of directed edges that are reciprocated

GlobalClusteringCoefficient fraction of length-two paths that are closed

### Homophily, Assortative Mixing, and Similarity

GraphAssortativity within-group connectivity minus between-group connectivity

VertexCorrelationSimilarity correlation similarity between actors