Just had this thought and wondered if there was a name for this type of graph that satisfies this transitivity property (would more easily let me read what other people have learned about it if so).
A weighted graph where given any vertices $v_i, v_j, v_k$, the weight of edge $$w(v_i, v_k) = w(v_i, v_j) + w(v_j, v_k)$$
Suppose $G$ has at least $3$ vertices, and satisfies the specified edge-weight conditions.
Let $a,b,c$ be distinct vertices of $G$, and let $$r=w(a,b),\;\;\;s=w(b,c),\;\;t=w(c,a)$$ Then we get the system of equations $$ \begin{cases} r=s+t\\[4pt] s=t+r\\[4pt] t=r+s\\ \end{cases} $$ which implies $r=s=t=0$.
Thus, all edge weights must be zero.