A labelling of the vertices of the Petersen graph with integers
The following labelling, whose sum is $37294$, improves Aaron Meyerowitz's by $64$. It was also found by extensive computer search, but with the strategy of assigning and fixing the first five primes (in all possible permutations) to the edges joining the vertices of the exterior and interior pentagons, and then examining the resulting sums of assigning all $10!$ possible permutations of the next ten primes to the remaining $10$ edges.
This by no means exhausts all possibilities, but does seem to come close to the optimal solution.
The above figure shows the initial assignment of the first five primes to the vertices of the graph.
The minimum value is $37294$ as described by F. Barrera.
I broke the symmetry a little by identifying $9$ inequivalent triples of edges to which the primes $\{41,43,47\}$ can be assigned, wrote a constraint satisfaction program for the problem, and then used Minion to solve it.
(I am sure there are more efficient ways to do this.)
With some computer searching I can get a sum of $37358$ and another of $37360.$ I list them below. I'm not saying they are optimal, though they seem pretty good.
The ten vertex labels should multiply to $N^2$ where $N$ is the product of the first $15$ primes. If they were allowed to be positive reals subject to this product then the minimum sum would come from setting them all to $N^{1/5} \approx 3612.1.$ Hence 36121 is a lower bound for the minimum possible sum.
The two examples are
$ [4879, 3913, 3182, 2162, 3995, 3441, 2337, 4807, 4147, 4495]=$ $ [ 7\cdot17\cdot41, 7\cdot13\cdot43, 2\cdot37\cdot43, 2\cdot23\cdot47, 5\cdot17\cdot47, $ $ 3\cdot31\cdot37, 3\cdot19\cdot41,11\cdot19\cdot23,11\cdot13\cdot29,5\cdot29\cdot31] $
with sum $37358$
And $[2337, 2726, 3055, 3182, 3565, 3731, 3999, 4301, 4403, 6061]=$ $[ 3\cdot 19\cdot 41 , 2\cdot 29\cdot 47 , 5\cdot 13\cdot 47 , 2\cdot 37\cdot 43 , 5\cdot 23\cdot 31 ,$ $ 7\cdot 13\cdot 41 , 3\cdot 31\cdot 43 , 11\cdot 17\cdot 23 , 7\cdot 17\cdot 37 , 11 \cdot 19\cdot 29 ] $
with sum $37360.$
These are each minimal under switching any two edge labels. checking all $2\binom{15}{3}$ cyclic three way switches before or after didn't improve anything.
The next smallest sums from $2000000$ random labellings followed by optimization were $37438, 37458, 37490, 37494$