# 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 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.

## 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** 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:

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

## See also

- Distance matrix
- Resistance distance
- Betweenness centrality
- Centrality
- Closeness
- Degree diameter problem for graphs and digraphs
- Metric graph

## 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

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