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 $d(u,v)$ between two vertices $u$ and $v$ is defined as the length of a shortest path from $u$ to $v$ consisting of arcs, provided at least one such path exists.^[3] Notice that, in contrast with the case of undirected graphs, $d(u,v)$ does not necessarily coincide with $d(v,u)$, 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 $\epsilon(v)$ 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 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 $u$.
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.

See also

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

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