# Is there an name for a graph with this property?

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 $$w(v_i, v_k) = w(v_i, v_j) + w(v_j, v_k)$$

• Does the existence of edges $(v_i,v_j)$ and $(v_j,v_k)$ mean that there must exist a $(v_i,v_k)$ with the weight you specify, or only that if it exists then its weight must be as given by your equation? – Henning Makholm Jul 3 '18 at 15:39

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 $r=s=t=0$.
• that is assuming $G$ is a complete undirected graph – gt6989b Jul 3 '18 at 16:21
• Yes, I'm assuming that for any two distinct vertices $a,b$, the edge $ab$ has some weight, and that $w(a,b)=w(b,a)$. – quasi Jul 3 '18 at 16:45