Which graph with an automorphism group isomorphic to the quaternion group $Q_8$ minimizes $|V|+3|E|$? In Symmetries of partial Latin squares, it is shown that for any graph $\Gamma=(V,E)$ with automorphism group $G$, there is a partial Latin square with $|V|+3|E|+49$ filled cells whose autotopism group is isomorphic to $G$.
Hence my question...

Which graph with an automorphism group isomorphic to the quaternion group $Q_8$ minimizes $|V|+3|E|$?

The answer to the question Smallest graph with automorphism group the quaternion $8$-group, $Q_8$ gives an example of a 16-vertex 48-edge graph with automorphism group $Q_8$.  It could be one that minimizes $|V|+3|E|$.
I'm looking to find partial Latin rectangles that (a) have symmetry groups isomorphic to a given group $G$, and (b) have low weight [number of filled cells].  The quaternion group is among the groups of smallest order I have yet to compute examples.  The answer to this question will give an upper bound on the weight.
 A: This answer is incomplete in that it only gives the upper bound $|V| + 3 |E| \leq 138$, which improves significantly on the bound of 160 given by the graph in the answer to the linked question.
That answer shows that any graph with automorphism group $Q_8$ has at least $16$ vertices. Now, this presentation from a 2016 REU by Edwards, Khanal, and Webster gives the minimum number $e(v)$ of edges among graphs on $v$ vertices with automorphism group $Q_8$ for the first several $v \geq 16$:
\begin{align*}
e(16) = e(17) &= 44 \\
e(18) = e(19) &= 40 \\
e(20)         &= 41 \\
\end{align*}
(This result doesn't appear to have been published otherwise.)
Among these the lowest value of $|V| + 3 |E|$ is achieved by a graph with $|V| = 18$, $|E| = 40$, for which $|V| + 3 |E| = 138$. Page 15 of the presentation gives an explicit example:

This leads to a partial Latin square with $138 + 49 = 187$ filled squares with autotopism group $Q_8$
In particular, if a graph with $v \geq 21$ vertices achieves a lower value of $|V| + 3 |E|$, it must have fewer than $\left\lceil 46 - \frac{v}{3}\right\rceil$ edges.
