# VertexOutDegree

gives the list of vertex out-degrees for all vertices in the graph g.

VertexOutDegree[g,v]

gives the vertex out-degree for the vertex v.

VertexOutDegree[{vw,},]

uses rules vw to specify the graph g.

# Details • The vertex out-degree for a vertex v is the number of outgoing directed edges from v.
• For an undirected graph g, an edge is taken to be both an in-edge and an out-edge. »

# Examples

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## Basic Examples(2)

Find the out-degree for each vertex:

Find the out-degree for a specified vertex:

## Scope(6)

VertexOutDegree works with directed graphs:

Undirected graphs:

Multigraphs:

Vertex out-degree for a vertex:

Use rules to specify the graph:

VertexOutDegree works with large graphs:

## Applications(4)

Highlight the vertex by its vertex out-degree for CycleGraph:

Show the out-degree histogram for :

The out-degree distribution follows BinomialDistribution[n-1,p]:

A PriceGraphDistribution is constructed, adding new vertices with a constant number of edges:

In this case, three edges from each new vertex (except initially) are added:

Create a food chain where an edge indicates what animals and insects eat:

The out-degree corresponds to the number of predators for the species:

Animals with zero out-degree are called top species or apex predators:

## Properties & Relations(7)

The out-degree of an undirected graph is the number of edges incident to each vertex:

Self-loops are counted twice:

Undirected graphs correspond to directed graphs with each edge both an in- and out-edge:

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 out-degrees of all vertices of an undirected graph is twice the number of edges:

The sum of the out-degrees of all vertices of a directed graph is equal to the number of edges:

A connected directed graph is Eulerian iff every vertex has equal in-degree and out-degree: