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 ^[1] because it is the length of the graph geodesic between those two vertices.^[2] If there is no path connecting the two vertices, i.e., if they belong to different connected components, then conventionally the distance is defined as infinite.

1 Related concepts
2 Algorithm for finding pseudo-peripheral vertices
3 See also
4 Notes

Related concepts

A metric space defined over a set of points in terms of distances in a graph defined over the set is called a graph metric. The vertex set (of an undirected graph) and the distance function form a metric space, if and only if the graph is connected.

The eccentricity $\epsilon(v)$ of a vertex $v$ is the greatest geodesic distance between $v$ and any other vertex. It can be thought of as how far a node is from the node most distant from it in the graph.

The radius $r$ of a graph is the minimum eccentricity of any vertex or, in symbols, $r = \min_{v \in V} \epsilon(v)$ .

The diameter $d$ of a graph is the maximum eccentricity of any vertex in the graph. That is, $d$ it is the greatest distance between any pair of vertices or, alternatively, $d = \max_{v \in V}\epsilon(v)$ . To find the diameter of a graph, first find the shortest path between each pair of vertices. The greatest length of any of these paths is the diameter of the graph.

A central vertex in a graph of radius $r$ is one whose eccentricity is $r$ —that is, a vertex that achieves the radius or, equivalently, a vertex $v$ such that $\epsilon(v) = r$ .

A peripheral vertex in a graph of diameter $d$ is one that is distance $d$ from some other vertex—that is, a vertex that achieves the diameter. Formally, $v$ is peripheral if $\epsilon(v) = d$ .

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, a vertex u is pseudo-peripheral, if for each vertex v with $d(u,v) = \epsilon(u)$ holds $\epsilon(u)=\epsilon(v)$ .

The partition of a graphs vertices into subsets by their distances from a given starting vertex is called the level structure of the graph.

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:

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

Notes

^ Bouttier, Jérémie; Di Francesco,P. ,Guitter, E. (July 2003). "Geodesic distance in planar graphs". Nuclear Physics B 663 (3): 535–567. doi:10.1016/S0550-3213(03)00355-9. Retrieved 2008-04-23. "By distance we mean here geodesic distance along the graph, namely the length of any shortest path between say two given faces"
^ Weisstein, Eric W.. "Graph Geodesic". MathWorld--A Wolfram Web Resource. Wolfram Research. Retrieved 2008-04-23. "The length of the graph geodesic between these points d(u,v) is called the graph distance between u and v"

Contents

Related concepts

Algorithm for finding pseudo-peripheral vertices

See also

Notes