---
title: "Graph Properties & Measurements"
language: "en"
type: "Guide"
summary: "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."
sections: 
  - 
    title: "Basic Properties"
    link: "null"
  - 
    title: "Structural Properties"
    link: "null"
  - 
    title: "Graph Isomorphism"
    link: "null"
  - 
    title: "Graph Coloring"
    link: "null"
  - 
    title: "Basic Measures"
    link: "null"
  - 
    title: "Distance Measures"
    link: "null"
  - 
    title: "Connectivity Measures"
    link: "null"
  - 
    title: "Centrality Measures"
    link: "null"
  - 
    title: "Reciprocity and Transitivity"
    link: "null"
  - 
    title: "Homophily, Assortative Mixing, and Similarity"
    link: "null"
keywords: 
- centrality measure
- PageRank
- HITS
- closeness
- betweenness
- closeness centrality
- betweenness centrality
- degree centrality
- eigenvector centrality
- Katz centrality
- degree
- in-degree
- out-degree
- number of vertices
- number of edges
- geodesic distance
- graph distance
- shortest path
canonical_url: "https://reference.wolfram.com/language/guide/GraphPropertiesAndMeasurements.html"
source: "Wolfram Language Documentation"
related_guides: 
  - 
    title: "Graphs & Networks"
    link: "https://reference.wolfram.com/language/guide/GraphsAndNetworks.en.md"
  - 
    title: "Paths, Cycles, and Flows"
    link: "https://reference.wolfram.com/language/guide/GraphPathsCyclesAndFlows.en.md"
  - 
    title: "Graphs and Matrices"
    link: "https://reference.wolfram.com/language/guide/GraphsAndMatrices.en.md"
  - 
    title: "Social Network Analysis"
    link: "https://reference.wolfram.com/language/guide/SocialNetworks.en.md"
related_functions: 
  - 
    title: "AcyclicGraphQ"
    link: "https://reference.wolfram.com/language/ref/AcyclicGraphQ.en.md"
  - 
    title: "BetweennessCentrality"
    link: "https://reference.wolfram.com/language/ref/BetweennessCentrality.en.md"
  - 
    title: "BipartiteGraphQ"
    link: "https://reference.wolfram.com/language/ref/BipartiteGraphQ.en.md"
  - 
    title: "CanonicalGraph"
    link: "https://reference.wolfram.com/language/ref/CanonicalGraph.en.md"
  - 
    title: "ChromaticPolynomial"
    link: "https://reference.wolfram.com/language/ref/ChromaticPolynomial.en.md"
  - 
    title: "ClosenessCentrality"
    link: "https://reference.wolfram.com/language/ref/ClosenessCentrality.en.md"
  - 
    title: "CompleteGraphQ"
    link: "https://reference.wolfram.com/language/ref/CompleteGraphQ.en.md"
  - 
    title: "ConnectedGraphQ"
    link: "https://reference.wolfram.com/language/ref/ConnectedGraphQ.en.md"
  - 
    title: "DegreeCentrality"
    link: "https://reference.wolfram.com/language/ref/DegreeCentrality.en.md"
  - 
    title: "DirectedGraphQ"
    link: "https://reference.wolfram.com/language/ref/DirectedGraphQ.en.md"
  - 
    title: "EdgeBetweennessCentrality"
    link: "https://reference.wolfram.com/language/ref/EdgeBetweennessCentrality.en.md"
  - 
    title: "EdgeChromaticNumber"
    link: "https://reference.wolfram.com/language/ref/EdgeChromaticNumber.en.md"
  - 
    title: "EdgeConnectivity"
    link: "https://reference.wolfram.com/language/ref/EdgeConnectivity.en.md"
  - 
    title: "EdgeCount"
    link: "https://reference.wolfram.com/language/ref/EdgeCount.en.md"
  - 
    title: "EdgeQ"
    link: "https://reference.wolfram.com/language/ref/EdgeQ.en.md"
  - 
    title: "EdgeTransitiveGraphQ"
    link: "https://reference.wolfram.com/language/ref/EdgeTransitiveGraphQ.en.md"
  - 
    title: "EigenvectorCentrality"
    link: "https://reference.wolfram.com/language/ref/EigenvectorCentrality.en.md"
  - 
    title: "EmptyGraphQ"
    link: "https://reference.wolfram.com/language/ref/EmptyGraphQ.en.md"
  - 
    title: "EulerianGraphQ"
    link: "https://reference.wolfram.com/language/ref/EulerianGraphQ.en.md"
  - 
    title: "FindEdgeColoring"
    link: "https://reference.wolfram.com/language/ref/FindEdgeColoring.en.md"
  - 
    title: "FindGraphIsomorphism"
    link: "https://reference.wolfram.com/language/ref/FindGraphIsomorphism.en.md"
  - 
    title: "FindIsomorphicSubgraph"
    link: "https://reference.wolfram.com/language/ref/FindIsomorphicSubgraph.en.md"
  - 
    title: "FindPlanarColoring"
    link: "https://reference.wolfram.com/language/ref/FindPlanarColoring.en.md"
  - 
    title: "FindSubgraphIsomorphism"
    link: "https://reference.wolfram.com/language/ref/FindSubgraphIsomorphism.en.md"
  - 
    title: "FindVertexColoring"
    link: "https://reference.wolfram.com/language/ref/FindVertexColoring.en.md"
  - 
    title: "GlobalClusteringCoefficient"
    link: "https://reference.wolfram.com/language/ref/GlobalClusteringCoefficient.en.md"
  - 
    title: "GraphAssortativity"
    link: "https://reference.wolfram.com/language/ref/GraphAssortativity.en.md"
  - 
    title: "GraphAutomorphismGroup"
    link: "https://reference.wolfram.com/language/ref/GraphAutomorphismGroup.en.md"
  - 
    title: "GraphDensity"
    link: "https://reference.wolfram.com/language/ref/GraphDensity.en.md"
  - 
    title: "GraphDiameter"
    link: "https://reference.wolfram.com/language/ref/GraphDiameter.en.md"
  - 
    title: "GraphDistance"
    link: "https://reference.wolfram.com/language/ref/GraphDistance.en.md"
  - 
    title: "GraphDistanceMatrix"
    link: "https://reference.wolfram.com/language/ref/GraphDistanceMatrix.en.md"
  - 
    title: "GraphHub"
    link: "https://reference.wolfram.com/language/ref/GraphHub.en.md"
  - 
    title: "GraphLinkEfficiency"
    link: "https://reference.wolfram.com/language/ref/GraphLinkEfficiency.en.md"
  - 
    title: "GraphQ"
    link: "https://reference.wolfram.com/language/ref/GraphQ.en.md"
  - 
    title: "GraphRadius"
    link: "https://reference.wolfram.com/language/ref/GraphRadius.en.md"
  - 
    title: "GraphReciprocity"
    link: "https://reference.wolfram.com/language/ref/GraphReciprocity.en.md"
  - 
    title: "HamiltonianGraphQ"
    link: "https://reference.wolfram.com/language/ref/HamiltonianGraphQ.en.md"
  - 
    title: "HITSCentrality"
    link: "https://reference.wolfram.com/language/ref/HITSCentrality.en.md"
  - 
    title: "IsomorphicGraphQ"
    link: "https://reference.wolfram.com/language/ref/IsomorphicGraphQ.en.md"
  - 
    title: "IsomorphicSubgraphQ"
    link: "https://reference.wolfram.com/language/ref/IsomorphicSubgraphQ.en.md"
  - 
    title: "KatzCentrality"
    link: "https://reference.wolfram.com/language/ref/KatzCentrality.en.md"
  - 
    title: "LinkRankCentrality"
    link: "https://reference.wolfram.com/language/ref/LinkRankCentrality.en.md"
  - 
    title: "LocalClusteringCoefficient"
    link: "https://reference.wolfram.com/language/ref/LocalClusteringCoefficient.en.md"
  - 
    title: "LoopFreeGraphQ"
    link: "https://reference.wolfram.com/language/ref/LoopFreeGraphQ.en.md"
  - 
    title: "MeanClusteringCoefficient"
    link: "https://reference.wolfram.com/language/ref/MeanClusteringCoefficient.en.md"
  - 
    title: "MeanDegreeConnectivity"
    link: "https://reference.wolfram.com/language/ref/MeanDegreeConnectivity.en.md"
  - 
    title: "MeanGraphDistance"
    link: "https://reference.wolfram.com/language/ref/MeanGraphDistance.en.md"
  - 
    title: "MeanNeighborDegree"
    link: "https://reference.wolfram.com/language/ref/MeanNeighborDegree.en.md"
  - 
    title: "MixedGraphQ"
    link: "https://reference.wolfram.com/language/ref/MixedGraphQ.en.md"
  - 
    title: "MultigraphQ"
    link: "https://reference.wolfram.com/language/ref/MultigraphQ.en.md"
  - 
    title: "PageRankCentrality"
    link: "https://reference.wolfram.com/language/ref/PageRankCentrality.en.md"
  - 
    title: "PathGraphQ"
    link: "https://reference.wolfram.com/language/ref/PathGraphQ.en.md"
  - 
    title: "PlanarGraphQ"
    link: "https://reference.wolfram.com/language/ref/PlanarGraphQ.en.md"
  - 
    title: "RadialityCentrality"
    link: "https://reference.wolfram.com/language/ref/RadialityCentrality.en.md"
  - 
    title: "SimpleGraphQ"
    link: "https://reference.wolfram.com/language/ref/SimpleGraphQ.en.md"
  - 
    title: "StatusCentrality"
    link: "https://reference.wolfram.com/language/ref/StatusCentrality.en.md"
  - 
    title: "TreeGraphQ"
    link: "https://reference.wolfram.com/language/ref/TreeGraphQ.en.md"
  - 
    title: "UndirectedGraphQ"
    link: "https://reference.wolfram.com/language/ref/UndirectedGraphQ.en.md"
  - 
    title: "VertexChromaticNumber"
    link: "https://reference.wolfram.com/language/ref/VertexChromaticNumber.en.md"
  - 
    title: "VertexConnectivity"
    link: "https://reference.wolfram.com/language/ref/VertexConnectivity.en.md"
  - 
    title: "VertexCorrelationSimilarity"
    link: "https://reference.wolfram.com/language/ref/VertexCorrelationSimilarity.en.md"
  - 
    title: "VertexCosineSimilarity"
    link: "https://reference.wolfram.com/language/ref/VertexCosineSimilarity.en.md"
  - 
    title: "VertexCount"
    link: "https://reference.wolfram.com/language/ref/VertexCount.en.md"
  - 
    title: "VertexDegree"
    link: "https://reference.wolfram.com/language/ref/VertexDegree.en.md"
  - 
    title: "VertexDiceSimilarity"
    link: "https://reference.wolfram.com/language/ref/VertexDiceSimilarity.en.md"
  - 
    title: "VertexEccentricity"
    link: "https://reference.wolfram.com/language/ref/VertexEccentricity.en.md"
  - 
    title: "VertexInDegree"
    link: "https://reference.wolfram.com/language/ref/VertexInDegree.en.md"
  - 
    title: "VertexJaccardSimilarity"
    link: "https://reference.wolfram.com/language/ref/VertexJaccardSimilarity.en.md"
  - 
    title: "VertexOutDegree"
    link: "https://reference.wolfram.com/language/ref/VertexOutDegree.en.md"
  - 
    title: "VertexQ"
    link: "https://reference.wolfram.com/language/ref/VertexQ.en.md"
  - 
    title: "VertexTransitiveGraphQ"
    link: "https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.en.md"
  - 
    title: "WeightedGraphQ"
    link: "https://reference.wolfram.com/language/ref/WeightedGraphQ.en.md"
---
# 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.

---

### Basic Properties

[`GraphQ`](https://reference.wolfram.com/language/ref/GraphQ.en.md) — test whether an expression is a graph object

[`DirectedGraphQ`](https://reference.wolfram.com/language/ref/DirectedGraphQ.en.md), [`UndirectedGraphQ`](https://reference.wolfram.com/language/ref/UndirectedGraphQ.en.md) — test whether a graph is directed or undirected

[`MultigraphQ`](https://reference.wolfram.com/language/ref/MultigraphQ.en.md), [`MixedGraphQ`](https://reference.wolfram.com/language/ref/MixedGraphQ.en.md) — test whether a graph is a multigraph or a mixed graph

* [`EdgeQ`](https://reference.wolfram.com/language/ref/EdgeQ.en.md)
* [`VertexQ`](https://reference.wolfram.com/language/ref/VertexQ.en.md)
* [`EmptyGraphQ`](https://reference.wolfram.com/language/ref/EmptyGraphQ.en.md)
* [`WeightedGraphQ`](https://reference.wolfram.com/language/ref/WeightedGraphQ.en.md)
* [`CompleteGraphQ`](https://reference.wolfram.com/language/ref/CompleteGraphQ.en.md)

### Structural Properties

[`SimpleGraphQ`](https://reference.wolfram.com/language/ref/SimpleGraphQ.en.md) — test whether a graph is simple

[`AcyclicGraphQ`](https://reference.wolfram.com/language/ref/AcyclicGraphQ.en.md) — test whether a graph is acyclic

* [`BipartiteGraphQ`](https://reference.wolfram.com/language/ref/BipartiteGraphQ.en.md)
* [`ConnectedGraphQ`](https://reference.wolfram.com/language/ref/ConnectedGraphQ.en.md)
* [`EulerianGraphQ`](https://reference.wolfram.com/language/ref/EulerianGraphQ.en.md)
* [`HamiltonianGraphQ`](https://reference.wolfram.com/language/ref/HamiltonianGraphQ.en.md)
* [`PathGraphQ`](https://reference.wolfram.com/language/ref/PathGraphQ.en.md)
* [`PlanarGraphQ`](https://reference.wolfram.com/language/ref/PlanarGraphQ.en.md)
* [`TreeGraphQ`](https://reference.wolfram.com/language/ref/TreeGraphQ.en.md)
* [`LoopFreeGraphQ`](https://reference.wolfram.com/language/ref/LoopFreeGraphQ.en.md)

### Graph Isomorphism

[`IsomorphicGraphQ`](https://reference.wolfram.com/language/ref/IsomorphicGraphQ.en.md) — test whether two graphs are the same after vertex renaming

[`FindGraphIsomorphism`](https://reference.wolfram.com/language/ref/FindGraphIsomorphism.en.md) — find the graph isomorphism as a list of rules

[`FindSubgraphIsomorphism`](https://reference.wolfram.com/language/ref/FindSubgraphIsomorphism.en.md) — find the subgraph isomorphism

* [`IsomorphicSubgraphQ`](https://reference.wolfram.com/language/ref/IsomorphicSubgraphQ.en.md)
* [`FindIsomorphicSubgraph`](https://reference.wolfram.com/language/ref/FindIsomorphicSubgraph.en.md)
* [`CanonicalGraph`](https://reference.wolfram.com/language/ref/CanonicalGraph.en.md)
* [`GraphAutomorphismGroup`](https://reference.wolfram.com/language/ref/GraphAutomorphismGroup.en.md)
* [`VertexTransitiveGraphQ`](https://reference.wolfram.com/language/ref/VertexTransitiveGraphQ.en.md)
* [`EdgeTransitiveGraphQ`](https://reference.wolfram.com/language/ref/EdgeTransitiveGraphQ.en.md)

### Graph Coloring

[`FindVertexColoring`](https://reference.wolfram.com/language/ref/FindVertexColoring.en.md) — find minimal vertex coloring

[`FindEdgeColoring`](https://reference.wolfram.com/language/ref/FindEdgeColoring.en.md) — find minimal edge coloring

[`FindPlanarColoring`](https://reference.wolfram.com/language/ref/FindPlanarColoring.en.md) — find face coloring for a planar graph layout

* [`VertexChromaticNumber`](https://reference.wolfram.com/language/ref/VertexChromaticNumber.en.md)
* [`EdgeChromaticNumber`](https://reference.wolfram.com/language/ref/EdgeChromaticNumber.en.md)
* [`ChromaticPolynomial`](https://reference.wolfram.com/language/ref/ChromaticPolynomial.en.md)

---

### Basic Measures

[`VertexCount`](https://reference.wolfram.com/language/ref/VertexCount.en.md), [`EdgeCount`](https://reference.wolfram.com/language/ref/EdgeCount.en.md) — give the number of vertices and edges in a graph

[`VertexDegree`](https://reference.wolfram.com/language/ref/VertexDegree.en.md) — give the number of edges for each vertex

* [`VertexInDegree`](https://reference.wolfram.com/language/ref/VertexInDegree.en.md)
* [`VertexOutDegree`](https://reference.wolfram.com/language/ref/VertexOutDegree.en.md)
* [`GraphHub`](https://reference.wolfram.com/language/ref/GraphHub.en.md)

### Distance Measures

[`GraphDistance`](https://reference.wolfram.com/language/ref/GraphDistance.en.md) — the length of the shortest path between two vertices

* [`MeanGraphDistance`](https://reference.wolfram.com/language/ref/MeanGraphDistance.en.md)
* [`GraphDistanceMatrix`](https://reference.wolfram.com/language/ref/GraphDistanceMatrix.en.md)
* [`VertexEccentricity`](https://reference.wolfram.com/language/ref/VertexEccentricity.en.md)
* [`GraphRadius`](https://reference.wolfram.com/language/ref/GraphRadius.en.md)
* [`GraphDiameter`](https://reference.wolfram.com/language/ref/GraphDiameter.en.md)

### Connectivity Measures

[`VertexConnectivity`](https://reference.wolfram.com/language/ref/VertexConnectivity.en.md) — the number of vertex-independent paths between two vertices

[`EdgeConnectivity`](https://reference.wolfram.com/language/ref/EdgeConnectivity.en.md) — the number of edge-independent paths between two vertices

[`GraphDensity`](https://reference.wolfram.com/language/ref/GraphDensity.en.md) — fraction of edges to the possible edges in a graph

[`GraphLinkEfficiency`](https://reference.wolfram.com/language/ref/GraphLinkEfficiency.en.md) — how tightly connected a graph is compared to number of edges

### Centrality Measures

[`ClosenessCentrality`](https://reference.wolfram.com/language/ref/ClosenessCentrality.en.md) — inverse average distance to every other vertex

[`BetweennessCentrality`](https://reference.wolfram.com/language/ref/BetweennessCentrality.en.md) — fraction of shortest paths that pass through the vertex

* [`DegreeCentrality`](https://reference.wolfram.com/language/ref/DegreeCentrality.en.md)
* [`EigenvectorCentrality`](https://reference.wolfram.com/language/ref/EigenvectorCentrality.en.md)
* [`KatzCentrality`](https://reference.wolfram.com/language/ref/KatzCentrality.en.md)
* [`PageRankCentrality`](https://reference.wolfram.com/language/ref/PageRankCentrality.en.md)
* [`HITSCentrality`](https://reference.wolfram.com/language/ref/HITSCentrality.en.md)
* [`RadialityCentrality`](https://reference.wolfram.com/language/ref/RadialityCentrality.en.md)
* [`StatusCentrality`](https://reference.wolfram.com/language/ref/StatusCentrality.en.md)
* [`EdgeBetweennessCentrality`](https://reference.wolfram.com/language/ref/EdgeBetweennessCentrality.en.md)
* [`LinkRankCentrality`](https://reference.wolfram.com/language/ref/LinkRankCentrality.en.md)

### Reciprocity and Transitivity

[`GraphReciprocity`](https://reference.wolfram.com/language/ref/GraphReciprocity.en.md) — fraction of directed edges that are reciprocated

[`GlobalClusteringCoefficient`](https://reference.wolfram.com/language/ref/GlobalClusteringCoefficient.en.md) — fraction of length-two paths that are closed

* [`MeanClusteringCoefficient`](https://reference.wolfram.com/language/ref/MeanClusteringCoefficient.en.md)
* [`LocalClusteringCoefficient`](https://reference.wolfram.com/language/ref/LocalClusteringCoefficient.en.md)

### Homophily, Assortative Mixing, and Similarity

[`GraphAssortativity`](https://reference.wolfram.com/language/ref/GraphAssortativity.en.md) — within-group connectivity minus between-group connectivity

[`VertexCorrelationSimilarity`](https://reference.wolfram.com/language/ref/VertexCorrelationSimilarity.en.md) — correlation similarity between actors

* [`MeanNeighborDegree`](https://reference.wolfram.com/language/ref/MeanNeighborDegree.en.md)
* [`MeanDegreeConnectivity`](https://reference.wolfram.com/language/ref/MeanDegreeConnectivity.en.md)
* [`VertexDiceSimilarity`](https://reference.wolfram.com/language/ref/VertexDiceSimilarity.en.md)
* [`VertexJaccardSimilarity`](https://reference.wolfram.com/language/ref/VertexJaccardSimilarity.en.md)
* [`VertexCosineSimilarity`](https://reference.wolfram.com/language/ref/VertexCosineSimilarity.en.md)

## Related Guides

* [Graphs & Networks](https://reference.wolfram.com/language/guide/GraphsAndNetworks.en.md)
* [Paths, Cycles, and Flows](https://reference.wolfram.com/language/guide/GraphPathsCyclesAndFlows.en.md)
* [Graphs and Matrices](https://reference.wolfram.com/language/guide/GraphsAndMatrices.en.md)
* [Social Network Analysis](https://reference.wolfram.com/language/guide/SocialNetworks.en.md)