TITLE: Polyhedral graphs from consecutive power distances
QUESTION [5 upvotes]: Raise a real value to powers 0 to $n-1$ for $n$ different lengths. Build a polyhedral graph from those lengths, none missing or repeated.  
I have found 4 solutions, all wheel graphs. Are there any others? In the images below, the number indicates the power.  The 0 edge has length $a^0=1$. 
 
These use the 2nd real values of the following polynomials:
$$x^6-x^2-1,\\ x^8-x^2-1, \\x^{12}+x^8-x^4-x^2-1, \\6 x^{16}+x^{15}-3 x^{14}-2 x^{13}-3 x^{11}+8 x^{10}-2 x^9-x^8-6 x^7-x^6+x^4-4 x^3-4 x^2+2 x+6$$
Are there any other solutions?

REPLY [3 votes]: Nautilus Graphs
There appear to be an infinite series of "Nautilus" wheel-graphs from order n=8 onward with an increasing number of "chambers". The choice of the name "Nautilus" becomes apparent for largish n. Eg:

I've generated Nautiluses up to order n=52, but higher orders require finding the root of polynomials of progressively higher degrees. I presume the series to be unbounded, but have not proven this. Inspiration for looking at these "Nautilus" graphs came from a Facebook post of an order n=14 graph by Ed Pegg.
Tetrahedral Graphs
If I've not made a mistake, there are six distinct, non-degenerate configurations for the tetrahedral graph:






Note that none of these conform to the "Nautilus" edge pattern.