Triameter (graph theory)

In graph theory, the triameter is a metric invariant that generalizes the concept of a graph's diameter. It is defined as the maximum sum of pairwise distances between any three vertices in a connected graph ${\textstyle G}$ and is denoted by

where ${\textstyle V}$ is the vertex set of ${\textstyle G}$ and ${\textstyle d(u,v)}$ is the length of the shortest path between vertices ${\textstyle u}$ and ${\textstyle v}$.

It extends the idea of the diameter, which captures the longest path between any two of its vertices. A triametral triple is a set of three vertices achieving ${\textstyle \mathop {\mathrm {tr} } (G)}$.