Is it really impossible to build such a bike? Given a strong enough material it would be possible. I have not idea about the structural analysis in this case. Does such a material exist?
Likely not impossible, but the crucial weak point would probably be the connection with the seat tube. Independent of the material, that place would need some heavy gusseting/reinforcement. That particular bike also has a very vulnerable connection to the head tube, as only the down tube connects to it.
There are bikes without a rear triangle, but they (almost?) always lack the seatstay instead of the chainstay or have what could be considered something in between a chainstay/seatstay. Here are some examples:
"I have not idea about the structural analysis in this case."
There's your problem. The chainstays are there for Good Reasons. http://www.mece.ualberta.ca/tutorials/ansys/BT/Bike/Bike.htm... If you take them out, now most forces on the bike effect one point in the frame, right beneath the saddle. Making the frame stronger requires more material (=weight) or better material (=price), whereas a rhombus of two triangles solves the problem efficiently.
In other words, you are proposing a problem to which the current bike design is the solution.
I can't see what the advantage would be though - if you do have a wonder material better to go with a more standard frame design and increase stiffness and/or reduce weight.
NB ~20 years ago I did have an "elevated stay" frame which was neat in some ways - no chain slap and the chain didn't go through the frame. However, the frame failed catastrophically while I was at traffic lights....
