The answers are that and that every vertex of has degree .
Proof:
The positive real solutions to form a curve of cardinality , and at most of these have or rational, so . (It cannot be larger, because this is also the number of pairs of vertices.)
For with , the set of positive real numbers such that is a prime number contains at most one rational number since if there were two, then we would obtain an equation with distinct primes and rational, which is impossible by unique factorization. Thus has infinite degree. On the other hand, the degree of is at most countable since and have at most one solution each for each . Finally, the same arguments apply when , with reciprocals of primes in place of primes.
Remark: The same statements hold for except that one should exclude the vertex .
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