WOLFRAM

Wolfram Language & System Documentation Center

GraphDistanceMatrix[g]

gives a matrix in which the (i,j)^(th) entry is the length of a shortest path in g between vertices i and j.

GraphDistanceMatrix[g,Parent]

returns a three-dimensional matrix in which the (1,i,j)^(th) entry is the length of a shortest path from i to j and the (2,i,j)^(th) entry is the predecessor of j in a shortest path from i to j.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
Method  
See Also
Tech Notes
Related Guides
Cite this Page
GraphUtilities`
GraphUtilities`

GraphDistanceMatrix

As of Version 10, all the functionality of the GraphUtilities package is built into the Wolfram System. »

GraphDistanceMatrix[g]

gives a matrix in which the (i,j)^(th) entry is the length of a shortest path in g between vertices i and j.

GraphDistanceMatrix[g,Parent]

returns a three-dimensional matrix in which the (1,i,j)^(th) entry is the length of a shortest path from i to j and the (2,i,j)^(th) entry is the predecessor of j in a shortest path from i to j.

Details and Options

Examples

open all close all

Basic Examples  (2)

Wolfram Language code: Needs["GraphUtilities`"]

This defines a simple directed graph:

Wolfram Language code: g = SparseArray[{{1, 2} -> 1., {2, 3} -> 1., {1, 5} -> 1., {4, 3} -> 1., {5, 4} -> -2.}, {5, 5}];
Wolfram Language code: GraphPlot[g, VertexLabeling -> True, EdgeRenderingFunction -> ({Arrow[#, 0.1], Text[g[[First[#2], Last[#2]]], LineScaledCoordinate[#1], Background -> White]}&)]

This calculates the distance between the vertices:

Wolfram Language code: GraphDistanceMatrix[g]//MatrixForm

This function has been superseded by GraphDistanceMatrix in the Wolfram System:

Wolfram Language code: g = WeightedAdjacencyGraph[SparseArray[{{1, 2} -> 1., {2, 3} -> 1., {1, 5} -> 1., {4, 3} -> 1., {5, 4} -> -2.}, {5, 5}, Infinity]]
Wolfram Language code: GraphDistanceMatrix[g]//MatrixForm

Scope  (1)

Wolfram Language code: Needs["GraphUtilities`"]

This defines a simple directed graph:

Wolfram Language code: g = SparseArray[{{1, 2} -> 1., {2, 3} -> 1., {1, 5} -> 1., {4, 3} -> 1., {5, 4} -> -2.}, {5, 5}];
Wolfram Language code: GraphPlot[g, VertexLabeling -> True, EdgeRenderingFunction -> ({Arrow[#, 0.1], Text[g[[First[#2], Last[#2]]], LineScaledCoordinate[#1], Background -> White]}&)]

This calculates the distance between the vertices:

Wolfram Language code: GraphDistanceMatrix[g]//MatrixForm

This shows also the predecessors in the shortest path:

Wolfram Language code: prd = GraphDistanceMatrix[g, Parent][[2]]; prd//MatrixForm

The path from 1 to 3 is {1,5,4,3}:

Wolfram Language code: Drop[FixedPointList[prd[[1, #]]&, 3], -1]//Reverse

This confirms the shortest path:

Wolfram Language code: GraphPath[g, 1, 3]

Options  (2)

Method  (2)

This defines a small graph:

Wolfram Language code: Needs["GraphUtilities`"]
Wolfram Language code: g0 = ToCombinatoricaGraph[{1 -> 2, 2 -> 3, 3 -> 1, 1 -> 3}]; g = SetEdgeWeights[g0, {{1, 2}, {2, 3}, {3, 1}, {1, 3}}, {-1, -2, 3, 1}]; adj = AdjacencyMatrix[g];
Wolfram Language code: GraphPlot[g0, MultiedgeStyle -> True, VertexLabeling -> True, EdgeRenderingFunction -> ({Arrow[#, 0.1], Text[adj[[First[#2], Last[#2]]], LineScaledCoordinate[#1], Background -> White]}&)]

Because of the negative edge weight, the Dijkstra algorithm cannot be applied:

Wolfram Language code: GraphDistanceMatrix[g, Method -> "Dijkstra"]

Both the Floyd–Warshall and Johnson algorithms work:

Wolfram Language code: GraphDistanceMatrix[g, Method -> "FloydWarshall"]
Wolfram Language code: GraphDistanceMatrix[g, Method -> "Johnson"]
Wolfram Language code: Needs["GraphUtilities`"]

This defines a small graph with a negative cycle:

Wolfram Language code: g = {{0, 5, 4, 0}, {0, 0, 3, -3}, {0, 0.0000001, 0, -1}, {-2, 0, 2, 0}};
Wolfram Language code: GraphPlot[{{1 -> 2, 5}, {1 -> 3, 4}, {2 -> 3, 3}, {2 -> 4, -3}, {3 -> 2, 0.0000001}, {3 -> 4, -1}, {4 -> 1, -2}, {4 -> 3, 2}}, VertexLabeling -> True, EdgeRenderingFunction -> ({Arrow[#, .05], Text[#3, LineScaledCoordinate[#1, .3], Background -> White]}&)]

The Dijkstra algorithm does not work for negative edge weights:

Wolfram Language code: GraphDistanceMatrix[g, Method -> "Dijkstra"]

The Floyd–Warshall algorithm does not detect any negative weight cycle, and gives the wrong answer:

Wolfram Language code: GraphDistanceMatrix[g, Method -> "FloydWarshall"]

The Johnson algorithm detects a negative weight cycle:

Wolfram Language code: GraphDistanceMatrix[g, Method -> "Johnson"]

The default algorithm for graphs with negative edge weights is Johnson:

Wolfram Language code: GraphDistanceMatrix[g]
Wolfram Research (2007), GraphDistanceMatrix, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/GraphDistanceMatrix.html.

Text

Wolfram Research (2007), GraphDistanceMatrix, Wolfram Language function, https://reference.wolfram.com/language/GraphUtilities/ref/GraphDistanceMatrix.html.

CMS

Wolfram Language. 2007. "GraphDistanceMatrix." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/GraphUtilities/ref/GraphDistanceMatrix.html.

APA

Wolfram Language. (2007). GraphDistanceMatrix. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/GraphUtilities/ref/GraphDistanceMatrix.html

BibTeX

@misc{reference.wolfram_2026_graphdistancematrix, author="Wolfram Research", title="{GraphDistanceMatrix}", year="2007", howpublished="\url{https://reference.wolfram.com/language/GraphUtilities/ref/GraphDistanceMatrix.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_graphdistancematrix, organization={Wolfram Research}, title={GraphDistanceMatrix}, year={2007}, url={https://reference.wolfram.com/language/GraphUtilities/ref/GraphDistanceMatrix.html}, note=[Accessed: 13-June-2026]}