Details and Options

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.

List of all options

AlignmentPoint	Center	the default point in the graphic to align with
AnnotationRules	{}	annotations for graph, edges and vertices
AspectRatio	Automatic	ratio of height to width
Axes	False	whether to draw axes
AxesLabel	None	axes labels
AxesOrigin	Automatic	where axes should cross
AxesStyle	{}	style specifications for the axes
Background	None	background color for the plot
BaselinePosition	Automatic	how to align with a surrounding text baseline
BaseStyle	{}	base style specifications for the graphic
ContentSelectable	Automatic	whether to allow contents to be selected
CoordinatesToolOptions	Automatic	detailed behavior of the coordinates tool
DirectedEdges	Automatic	whether to interpret Rule as DirectedEdge
EdgeLabels	None	labels and label placements for edges
EdgeLabelStyle	Automatic	style to use for edge labels
EdgeShapeFunction	Automatic	generate graphic shapes for edges
EdgeStyle	Automatic	style used for edges
EdgeWeight	Automatic	weights for edges
Epilog	{}	primitives rendered after the main plot
FormatType	TraditionalForm	the default format type for text
Frame	False	whether to put a frame around the plot
FrameLabel	None	frame labels
FrameStyle	{}	style specifications for the frame
FrameTicks	Automatic	frame ticks
FrameTicksStyle	{}	style specifications for frame ticks
GraphHighlight	{}	graph elements to highlight
GraphHighlightStyle	Automatic	style for highlight
GraphLayout	Automatic	how to lay out vertices and edges
GridLines	None	grid lines to draw
GridLinesStyle	{}	style specifications for grid lines
ImageMargins	0.	the margins to leave around the graphic
ImagePadding	All	what extra padding to allow for labels etc.
ImageSize	Automatic	the absolute size at which to render the graphic
LabelStyle	{}	style specifications for labels
Method	Automatic	details of graphics methods to use
PerformanceGoal	Automatic	aspects of performance to try to optimize
PlotLabel	None	an overall label for the plot
PlotRange	All	range of values to include
PlotRangeClipping	False	whether to clip at the plot range
PlotRangePadding	Automatic	how much to pad the range of values
PlotRegion	Automatic	the final display region to be filled
PlotTheme	$PlotTheme	overall theme for the graph
PreserveImageOptions	Automatic	whether to preserve image options when displaying new versions of the same graphic
Prolog	{}	primitives rendered before the main plot
RotateLabel	True	whether to rotate y labels on the frame
Ticks	Automatic	axes ticks
TicksStyle	{}	style specifications for axes ticks
VertexCoordinates	Automatic	coordinates for vertices
VertexLabels	None	labels and placements for vertices
VertexLabelStyle	Automatic	style to use for vertex labels
VertexShape	Automatic	graphic shape for vertices
VertexShapeFunction	Automatic	generate graphic shapes for vertices
VertexSize	Medium	size of vertices
VertexStyle	Automatic	styles for vertices
VertexWeight	Automatic	weights for vertices

Examples

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Basic Examples (3)

Join two graphs:

Wolfram Language code: GraphJoin[[image], [image], ...]

Plot the structure of the adjacency matrices:

Wolfram Language code: GraphJoin @@@ Table[RandomGraph[{5, 7}], 3, 2]

Wolfram Language code: Map[MatrixPlot @* AdjacencyMatrix, %]

Generate wheel graphs:

Wolfram Language code: GraphJoin[Graph[{1}, {}], CycleGraph[14]]

Complete graphs :

Wolfram Language code: GraphJoin[Graph[Range[3], {}], Graph[Range[4], {}]]

Generate dipyramid graphs:

Wolfram Language code: GraphJoin[Graph[{1, 2}, {}], CycleGraph[6]]