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EdgeConnectivity[g]

gives the edge connectivity of the graph g.

EdgeConnectivity[g,s,t]

gives the s-t edge connectivity of the graph g.

EdgeConnectivity[{vw,…},…]

uses rules vw to specify the graph g.

Details and Options
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Options  
EdgeWeight  
Properties & Relations  
See Also
Related Guides
History
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EdgeConnectivity

EdgeConnectivity[g]

gives the edge connectivity of the graph g.

EdgeConnectivity[g,s,t]

gives the s-t edge connectivity of the graph g.

EdgeConnectivity[{vw,…},…]

uses rules vw to specify the graph g.

Details and Options

  • EdgeConnectivity is also known as line connectivity.
  • The edge connectivity of a graph g is the smallest number of edges whose deletion from g disconnects g.
  • The s-t edge connectivity is the smallest number of edges whose deletion from g disconnects g, with s and t in two different connected components.
  • For weighted graphs, EdgeConnectivity gives the smallest sum of edge weights.
  • For a disconnected graph, EdgeConnectivity will return 0.
  • The following option can be given:
  • EdgeWeight Automaticedge weight for each edge

Examples

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Basic Examples  (2)

Find the edge connectivity:

Wolfram Language code: EdgeConnectivity[[image]]

Find the edge connectivity between two vertices:

Wolfram Language code: EdgeConnectivity[[image], 2, 6]

Scope  (7)

EdgeConnectivity works on undirected graphs:

Wolfram Language code: EdgeConnectivity[[image]]

Directed graphs:

Wolfram Language code: EdgeConnectivity[[image]]

Weighted graphs:

Wolfram Language code: EdgeConnectivity[[image]]

Multigraphs:

Wolfram Language code: EdgeConnectivity[[image]]

Mixed graphs:

Wolfram Language code: EdgeConnectivity[[image]]

Use rules to specify the graph:

Wolfram Language code: EdgeConnectivity[{1 -> 2, 2 -> 3, 4 -> 3, 6 -> 1, 6 -> 5, 5 -> 2, 5 -> 4, 2 -> 6, 3 -> 5}]

EdgeConnectivity works on large graphs:

Wolfram Language code: g = GridGraph[{10, 10, 10, 10}];
Wolfram Language code: EdgeConnectivity[g]//Timing

Options  (1)

EdgeWeight  (1)

By default, the edge weight of an edge is taken to be its EdgeWeight property if available, otherwise 1:

Wolfram Language code: EdgeConnectivity[[image]]

Use EdgeWeight->weights to set the edge weight:

Wolfram Language code: EdgeConnectivity[[image], EdgeWeight -> Range[5]]

Properties & Relations  (3)

Use FindEdgeCut to compute the edge connectivity:

Wolfram Language code: g = [image];
Wolfram Language code: {Length[FindEdgeCut[g]], EdgeConnectivity[g]}

The maximum flow between two vertices is equal to the edge connectivity:

Wolfram Language code: g = [image];
Wolfram Language code: {FindMaximumFlow[g, 1, 2], EdgeConnectivity[g, 1, 2]}

EdgeConnectivity returns 0 for a disconnected graph:

Wolfram Language code: g = [image];
Wolfram Language code: ConnectedGraphQ[g]
Wolfram Language code: EdgeConnectivity[g]
Wolfram Research (2012), EdgeConnectivity, Wolfram Language function, https://reference.wolfram.com/language/ref/EdgeConnectivity.html (updated 2015).

Text

Wolfram Research (2012), EdgeConnectivity, Wolfram Language function, https://reference.wolfram.com/language/ref/EdgeConnectivity.html (updated 2015).

CMS

Wolfram Language. 2012. "EdgeConnectivity." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2015. https://reference.wolfram.com/language/ref/EdgeConnectivity.html.

APA

Wolfram Language. (2012). EdgeConnectivity. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/EdgeConnectivity.html

BibTeX

@misc{reference.wolfram_2026_edgeconnectivity, author="Wolfram Research", title="{EdgeConnectivity}", year="2015", howpublished="\url{https://reference.wolfram.com/language/ref/EdgeConnectivity.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_edgeconnectivity, organization={Wolfram Research}, title={EdgeConnectivity}, year={2015}, url={https://reference.wolfram.com/language/ref/EdgeConnectivity.html}, note=[Accessed: 13-June-2026]}