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 (also called a graph geodesic) connecting them. This is also known as the geodesic distance.[1] Notice that there may be more than one shortest path between 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.

In the case of a directed graph the distance between two vertices and is defined as the length of a shortest directed path from to consisting of arcs, provided at least one such path exists.[3] Notice that, in contrast with the case of undirected graphs, does not necessarily coincide with , and it might be the case that one is defined while the other is not.

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 of a vertex is the greatest geodesic distance between 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 of a graph is the minimum eccentricity of any vertex or, in symbols, .

The diameter of a graph is the maximum eccentricity of any vertex in the graph. That is, is the greatest distance between any pair of vertices or, alternatively, . 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 is one whose eccentricity is —that is, a vertex that achieves the radius or, equivalently, a vertex such that .

A peripheral vertex in a graph of diameter is one that is distance from some other vertex—that is, a vertex that achieves the diameter. Formally, is peripheral if .

A pseudo-peripheral vertex has the property that for any vertex , if is as far away from as possible, then is as far away from as possible. Formally, a vertex u is pseudo-peripheral, if for each vertex v with holds .

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

A graph such that for every pair of vertices there is a unique shortest path connecting them is called a geodetic graph. For example, all trees are geodetic.[4]

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 .
  2. Among all the vertices that are as far from as possible, let be one with minimal degree.
  3. If then set and repeat with step 2, else is a pseudo-peripheral vertex.

See also

Notes

  1. ↑ 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
  2. ↑ 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
  3. ↑ F. Harary, Graph Theory, Addison-Wesley, 1969, p.199.
  4. ↑ Øystein Ore, Theory of graphs [3rd ed., 1967], Colloquium Publications, American Mathematical Society, p. 104
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