This is documentation for Mathematica 8, which was
based on an earlier version of the Wolfram Language.

# VertexDegree

 VertexDegree[g] gives the list of vertex degrees for all vertices in the graph g. VertexDegreegives the vertex degree for the vertex v.
• The vertex degree for a vertex v is the number of edges incident to v.
• For a directed graph g, an edge is incident to a vertex whether it is an in-edge or an out-edge.
Find the degree for each vertex:
Find the degree for a specified vertex:
Vertex degrees of the HararyGraph:
Find the degree for each vertex:
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Find the degree for a specified vertex:
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Vertex degrees of the HararyGraph:
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 Scope   (4)
VertexDegree works with undirected graphs:
Directed graphs:
Vertex degree for a vertex:
Works with large graphs:
 Applications   (4)
Highlight the vertex by its vertex degree for CycleGraph:
An unbalanced tree:
Create a social network:
Find the people with more influence:
The degree distribution for a Bernoulli random graph follows a BinomialDistribution:
Generate vertex degrees from 1000 instances of random graphs:
Find the probability that a Bernoulli random graph has max degree greater than 50:
The vertex degree distribution for BarabasiAlbertGraphDistribution follows a power-law:
The degree of a vertex of an undirected graph is the number of edges incident to the vertex:
Self-loops are counted twice:
For an undirected graph, the vertex in-degree and out-degree are equal to the vertex degree:
For a directed graph, the sum of the vertex in-degree and out-degree is the vertex degree:
Put the vertex degree, in-degree, and out-degree before, above, and below the vertex, respectively:
The sum of the degrees of all vertices of a graph is twice the number of edges:
Every graph has an even number of vertices with odd degree:
Connected simple graphs have minimum vertex degree of at least :
A graph with minimum vertex degree at least 2 contains a cycle:
The vertex degrees of an undirected graph can be obtained from its adjacency matrix:
The vertex degrees of a directed graph can be obtained from its adjacency matrix:
The vertex degrees for an undirected graph can be obtained from the incidence matrix:
The vertex degrees for a directed graph can be obtained from the incidence matrix:
Each vertex of a -regular graph has the same vertex degree :
All vertices of a simple graph have maximum degree less than the number of vertices:
A simple graph without isolated vertices has at least one pair of vertices with equal degrees:
A connected undirected graph is Eulerian iff every vertex has an even degree:
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