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ConnectedGraphQ
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ConnectedGraphQ

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yields True if the graph g is connected, and False otherwise.

Details

  • ConnectedGraphQ works for any graph object.
  • A graph is connected if there is a path between every pair of vertices.

Examples

open allclose all

Basic Examples  (2)Summary of the most common use cases

Test whether a graph is connected:

In[1]:=1
✖
https://wolfram.com/xid/01yrpa3dx6b-rb3d4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01yrpa3dx6b-78jol
Out[2]=2

A graph with isolated vertices is not connected:

In[1]:=1
✖
https://wolfram.com/xid/01yrpa3dx6b-jfp3fm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01yrpa3dx6b-dxll7e
Out[2]=2

Scope  (6)Survey of the scope of standard use cases

Test undirected graphs:

In[1]:=1
✖
https://wolfram.com/xid/01yrpa3dx6b-bu4i36
Out[1]=1

Directed graphs:

In[1]:=1
✖
https://wolfram.com/xid/01yrpa3dx6b-idswig
Out[1]=1

Multigraphs:

In[1]:=1
✖
https://wolfram.com/xid/01yrpa3dx6b-5c4td0
Out[1]=1

Mixed graphs:

In[1]:=1
✖
https://wolfram.com/xid/01yrpa3dx6b-26nv2e
Out[1]=1

ConnectedGraphQ gives False for anything that is not a connected graph:

In[1]:=1
✖
https://wolfram.com/xid/01yrpa3dx6b-3l2bwe
Out[1]=1

ConnectedGraphQ works with large graphs:

In[1]:=1
✖
https://wolfram.com/xid/01yrpa3dx6b-do4pmf
In[2]:=2
✖
https://wolfram.com/xid/01yrpa3dx6b-3y4dmr
Out[2]=2

Applications  (1)Sample problems that can be solved with this function

Compute the probability that the Watts–Strogatz random graph model is connected:

In[1]:=1
✖
https://wolfram.com/xid/01yrpa3dx6b-fq4y0z
In[2]:=2
✖
https://wolfram.com/xid/01yrpa3dx6b-yjr7tg
In[3]:=3
✖
https://wolfram.com/xid/01yrpa3dx6b-n45h2i
Out[3]=3

Properties & Relations  (5)Properties of the function, and connections to other functions

The graph distance matrix of a connected graph does not have entries:

In[1]:=1
✖
https://wolfram.com/xid/01yrpa3dx6b-dlq5yi
Out[1]=1

Connected graph:

In[2]:=2
✖
https://wolfram.com/xid/01yrpa3dx6b-b4jw5q

Disconnected graph:

In[3]:=3
✖
https://wolfram.com/xid/01yrpa3dx6b-dkfpm0
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/01yrpa3dx6b-ewvtp2

The minimum number of edges in a connected graph with vertices is :

In[1]:=1
✖
https://wolfram.com/xid/01yrpa3dx6b-bge9sr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01yrpa3dx6b-bkwc58
Out[2]=2

A path graph with vertices has exactly edges:

In[3]:=3
✖
https://wolfram.com/xid/01yrpa3dx6b-iqoccl
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/01yrpa3dx6b-tfe5e
Out[4]=4

The sum of the vertex degrees of a connected graph is greater than for the underlying simple graph:

In[1]:=1
✖
https://wolfram.com/xid/01yrpa3dx6b-c9u2no
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01yrpa3dx6b-cb1njg
Out[2]=2

A disconnected graph:

In[3]:=3
✖
https://wolfram.com/xid/01yrpa3dx6b-ecasd1
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/01yrpa3dx6b-b7cnwi
Out[4]=4

An undirected tree is connected:

In[1]:=1
✖
https://wolfram.com/xid/01yrpa3dx6b-edtvky
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01yrpa3dx6b-e4qbaq
Out[2]=2

An undirected path is connected:

In[1]:=1
✖
https://wolfram.com/xid/01yrpa3dx6b-chaumv
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/01yrpa3dx6b-m3y6z
Out[2]=2
Wolfram Research (2010), ConnectedGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/ConnectedGraphQ.html.
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Wolfram Research (2010), ConnectedGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/ConnectedGraphQ.html.

Text

Wolfram Research (2010), ConnectedGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/ConnectedGraphQ.html.

✖
Wolfram Research (2010), ConnectedGraphQ, Wolfram Language function, https://reference.wolfram.com/language/ref/ConnectedGraphQ.html.

CMS

Wolfram Language. 2010. "ConnectedGraphQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ConnectedGraphQ.html.

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Wolfram Language. 2010. "ConnectedGraphQ." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/ConnectedGraphQ.html.

APA

Wolfram Language. (2010). ConnectedGraphQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ConnectedGraphQ.html

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Wolfram Language. (2010). ConnectedGraphQ. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/ConnectedGraphQ.html

BibTeX

@misc{reference.wolfram_2025_connectedgraphq, author="Wolfram Research", title="{ConnectedGraphQ}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/ConnectedGraphQ.html}", note=[Accessed: 29-May-2025 ]}

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@misc{reference.wolfram_2025_connectedgraphq, author="Wolfram Research", title="{ConnectedGraphQ}", year="2010", howpublished="\url{https://reference.wolfram.com/language/ref/ConnectedGraphQ.html}", note=[Accessed: 29-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_connectedgraphq, organization={Wolfram Research}, title={ConnectedGraphQ}, year={2010}, url={https://reference.wolfram.com/language/ref/ConnectedGraphQ.html}, note=[Accessed: 29-May-2025 ]}

✖
@online{reference.wolfram_2025_connectedgraphq, organization={Wolfram Research}, title={ConnectedGraphQ}, year={2010}, url={https://reference.wolfram.com/language/ref/ConnectedGraphQ.html}, note=[Accessed: 29-May-2025 ]}