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.)
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.