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Given a graphs $G$ with vertex set $V$ and edge set $E$, I am defining a new graph $M(E)$ whose vertex set is $E$ and I define two vertices in $M(E)$ are adjacent if they share a common vertex in $G$. is there any name for this graph associated to the given graph $G$ ?

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    @daw I cant find any reference in the name edge-dual graph. do you know any reference? thanks. – 2017-02-16
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    it's called the 'line graph', although this name is quite annoying to google for ... http://mathworld.wolfram.com/LineGraph.html – 2017-02-16
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    Thanks Gillian. I just found from my friends also. – 2017-02-16

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My friend just told me about this : The line graph L (G) of a graph G is what we are talking about whose definition can be found here : https://en.m.wikipedia.org/wiki/Line_graph