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What is an intersection graph? And how can I construct the intersection graph of the Desargues Configuration?

I'm currently trying to show that the intersection graph of the Desargues Configuration is isomorphic to another graph, but I want to visualize both graphs before proceeding.

(Note: The Desargues Configuration is a configuration of 10 points and ten lines, with three points per line and three lines per point. Refer to https://en.wikipedia.org/wiki/Desargues_configuration for further information)

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    A vertex for each line of the configuration, adjacent if the lines intersect. It's relatively simple, only 10 vertices so it should be simple to draw and investigate. – 2017-02-27
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    It might help many Readers to see a picture of the Desargues configuration as [the Wikipedia article](https://en.wikipedia.org/wiki/Desargues_configuration) illustrates. – 2017-02-27

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As stated in the comment by Morgan Rodgers, the lines in Desargues Configuration are the vertices in your intersection graph, and adjacency is defined by intersection of lines. So it should end up as a 6-regular graph on 10 vertices.