That's a decisive question. In other words, i think if you were looking for the web 2.0 look, i would say no because, the idea is to "brand" the name first, and brand the icon second.
However, if you were to add an icon, it's the symbol itself, that in long terms should convey the name and the brand, for example, the nike swoosh, no longer needs the "nike" type of "thinking"
My point is that it's the name that you want to "brand" not the symbol.
Perhaps as a suggestion, you may want to go in a new direction and see some designs submitted to creating the entire name as a wordmark,
I hope i've been some help, i will continue to work and come up with ideas in the coming days.
regards
Spiro