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How can the display of graphs using GraphData be customized?

For example, how can we customize the vertex style, color, edge thickness of the output toGraphData["TetrahedralGraph"]?

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Use GraphPlot and the many options there. Specifically for your question the options are VertexRenderingFunction and EdgeRenderingFunction. Example:

GraphPlot[GraphData["TetrahedralGraph"], 
 EdgeRenderingFunction -> ({Red, Thickness[.01], Line[#1]} &),
 VertexRenderingFunction -> ({Blue, PointSize[.03], Point[#1]} &)
]
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    $\begingroup$ I would recommend using SetProperty and the new Graph objects (since v8) instead of the deprecated GraphPlot. $\endgroup$ – Szabolcs Sep 9 '15 at 8:29

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