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Is there a conventional name for the resultant graph (H) obtained by deleting all loops and multiple edges from the original graph (G)?

Something along the lines of "Let H be the simple graph of G.." or "Let H be the simplified version of G.." where I would like to replace the boldfaced. (The former doesn't sound quite correct, while the latter is worse.)

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up vote 2 down vote accepted

I'm not sure if this is universal, but $H$ is sometimes referred to as the underlying simple graph of $G$. For example, this terminology is used in Bondy and Murty. An analogous term is often used for digraphs when referring to the underlying undirected graph of a digraph.

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Thanks for your heads-up. Sounds better than my suggestions already. – Ryan Apr 9 '13 at 17:08

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