Suppose that a new graph is built from two given unlabelled graphs $G$, $H$ with distinguished edges $e\in G$ and $f\in H$ by taking a copy of each graph and identifying $e$ with $f$ (and hence the vertices at their ends).
What would be a good notation for the resulting graph? And what would be a good notation if the edges do not need to be specified because the resulting construct is independent of the choice, e.g. $C_4$ and $C_5$?
I would use something like $$G\stackrel{e,f}\bowtie H \quad \text{ or } \quad G\stackrel{e\sim f}\sqcup H,$$ of course, without the upper part if it doesn't matter.
I hope this helps $\ddot\smile$