The term path metric is used for a length metric, in which distances are infima
of the lengths of continuous paths
joining the two points, with lengths measured using that
metric (Birtea et al. 2008, p. 466).
In graph theory, the name is also used for the graph distance on the vertex
set of a finite connected undirected graph (Geller et al. 2024, p. 5).
This counts the number of edges in a shortest graph
path between distinct vertices, with distance
0 from a vertex to itself. The vertex
set with this metric must be distinguished from the
realization obtained by replacing each edge by a closed interval of length
1 and equipping the result with the induced length metric
(Bridson and Haefliger 1999, pp. 6-7).
See also
Graph Distance,
Graph Metric,
Length Metric,
Length
Space,
Metric
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References
Birtea, P.; Ortega, J.-P.; and Ratiu, T. S. "A Local-to-Global Principle for Convexity in Metric Spaces." J. Lie Theory 18,
445-469, 2008. https://doi.org/10.5802/jolt.505.Bridson,
M. R. and Haefliger, A. Metric
Spaces of Non-Positive Curvature. Berlin, Germany: Springer-Verlag, 1999.
https://doi.org/10.1007/978-3-662-12494-9.Geller,
W.; Misiurewicz, M.; and Sawicki, D. "Coarse Entropy of Metric Spaces."
Geom. Dedicata 218, Art. 99, 2024. https://doi.org/10.1007/s10711-024-00925-z.
Cite this as:
Weisstein, Eric W. "Path Metric." From
MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PathMetric.html
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