Distance (graph theory)

In the mathematical field of graph theory, the distance between two vertices in a graph is the number of edges in a shortest path connecting them. This is also known as the geodesic distance.

There are a number of other measurements defined in terms of distance:

The eccentricity $$\epsilon$$ of a vertex $$v$$ is the greatest distance between $$v$$ and any other vertex.

The radius of a graph is the minimum eccentricity of any vertex.

The diameter of a graph is the maximum eccentricity of any vertex in the graph. That is, it is the greatest distance between any two vertices. A peripheral vertex in a graph of diameter $$d$$ is one that is distance $$d$$ from some other vertex&mdash;that is, a vertex that achieves the diameter.

A pseudo-peripheral vertex $$v$$ has the property that for any vertex $$u$$, if $$v$$ is as far away from $$u$$ as possible, then $$u$$ is as far away from $$v$$ as possible. Formally, if the distance from $$u$$ to $$v$$ equals the eccentricity of $$u$$, then it equals the eccentricity of $$v$$.

Algorithm for finding pseudo-peripheral vertices
Often peripheral sparse matrix algorithms need a starting vertex with a high eccentricity. A peripheral vertex would be perfect, but is often hard to calculate. In most circumstances a pseudo-peripheral vertex can be used. A pseudo-peripheral vertex can easily be found with the following algorithm:


 * 1) Choose a vertex $$u$$.
 * 2) Among all the vertices that are as far from $$u$$ as possible, let $$v$$ be one with minimal degree.
 * 3) If $$\epsilon(v) > \epsilon(u)$$ then set u=v and repeat with step 2, else v is a pseudo-peripheral vertex.