# GraphJoin

GraphJoin[g1,g2]

gives the graph join of the graphs g1 and g2.

# Details • GraphJoin is also known as complete union.
• GraphJoin is typically used to produce new graphs with particular reachable subgraphs.
• GraphJoin[g1,g2] gives a graph obtained from g1 and g2 by joining each vertex of g1 to all vertices of g2.
• • GraphJoin[g1,g2] returns GraphDisjointUnion[g1,g2] together with all edges between each vertex of IndexGraph[g1] and each vertex of IndexGraph[g2,n+1], where n is the VertexCount for g1.
• GraphJoin takes the same options as Graph.

# Examples

open allclose all

## Basic Examples(3)

Join two graphs:

Plot the structure of the adjacency matrices:

Generate wheel graphs:

Complete graphs :

Generate dipyramid graphs:

## Scope(29)

### Directed Graphs(5)

GraphJoin works with directed graphs:

Simple directed graphs:

Directed multigraphs:

Directed weighted graphs:

Directed annotated graphs:

### Undirected Graphs(5)

GraphJoin works with undirected graphs:

Simple undirected graphs:

Undirected multigraphs:

Undirected weighted graphs:

Undirected annotated graphs:

### Mixed Graphs(5)

GraphJoin works with mixed graphs:

Simple mixed graphs:

Mixed multigraphs:

Mixed weighted graphs:

Mixed annotated graphs:

### Multigraphs(5)

GraphJoin works with multigraphs:

Directed multigraphs:

Mixed multigraphs:

Weighted multigraphs:

Annotated multigraphs:

### Weighted Graphs(5)

GraphJoin works with weighted graphs:

Directed weighted graphs:

Undirected weighted graphs:

Mixed weighted graphs:

Annotated weighted graphs:

### Special Graphs(4)

GraphJoin works on entity graphs:

GraphJoin works on trees:

Use rules to specify the graph:

GraphJoin works with more than two graphs:

## Properties & Relations(4)

Joining the adjacency matrix of two graphs on diagonal and others are ones gets the adjacency matrix of GraphJoin:

The join of a single vertex to an empty graph gets a star graph:

The join of a single vertex to a cycle graph gets a wheel graph :

The join of empty graphs on , , ... nodes gets a complete -partite graph :