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distance (in a graph) (Definition)

The distance $ d(x,y)$ of two vertices $ x$ and $ y$ of a graph $ G$ is the length of the shortest path (or, equivalently, walk) from $ x$ to $ y$. If there is no path from $ x$ to $ y$ (i.e. if they lie in different components of G), we set $ d(x,y) := \infty.$

Two basic graph invariants involving distance are the diameter $ {\mathrm{diam}}G := \max_{(x,y)\in V(G)^2} d(x,y)$ (the maximum distance between two vertices of $ G$) and the radius $ {\mathrm{rad}}G := \min_{x\in V(G)} \max_{y\in V(G)} d(x,y)$ (the maximum distance of a vertex from a central vertex of $ G$, i.e. a vertex such that the maximum distance to another vertex is minimal).



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"distance (in a graph)" is owned by Cosmin. [ full author list (2) | owner history (1) ]
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See Also: graph, path, diameter, path

Other names:  distance
Also defines:  diameter (of a graph), radius (of a graph), central vertex
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Cross-references: minimal, radius, diameter, invariants, components, walk, path, length, graph, vertices
There are 12 references to this entry.

This is version 8 of distance (in a graph), born on 2002-03-07, modified 2007-10-08.
Object id is 2765, canonical name is DistanceInAGraph.
Accessed 9266 times total.

Classification:
AMS MSC: 05C12 (Combinatorics :: Graph theory :: Distance in graphs)
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