Alphabeta MathHow statement and proof provenance work
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The path metric of a connected simple graph is a metric on its vertex set
Statement
The path metric of a connected simple graph is a metric on its vertex set.
Facts & Assumptions
Given: The hypotheses of the Statement.
The path metric of a connected simple graph assigns to two vertices the least length of a path joining them (The path metric of a connected simple graph).
A walk of length in a simple graph is a finite vertex list with consecutive vertices adjacent; a path is a walk with distinct vertices; the graph is connected when it is nonempty and every two vertices are joined by a path (Walks, paths, connectedness and components in a simple graph on an arbitrary vertex set).
Every walk in a simple graph contains a path with the same endpoints and of no greater length (Every walk contains a path between the same endpoints, of no greater length).
A metric on a set satisfies separation, symmetry and the triangle inequality (Metric space: iff , symmetry, and the triangle inequality; pseudometric and ultrametric).
A simple graph is a pair with any set and a set of two-element subsets of (Simple graphs on an arbitrary vertex set).
Proof
The only path of length zero joins a vertex to itself, so the distance vanishes exactly on the diagonal.
Reversing a path preserves its length, so the distance is symmetric.
Concatenating two shortest paths gives a walk of the summed length, which the previous lemma replaces by a path no longer, giving the triangle inequality.
Steps 1.1, 1.2 and 1.3 establish separation, symmetry and the triangle inequality, so the path metric is a metric.
Depends on
- Simple graphs on an arbitrary vertex set
- Walks, paths, connectedness and components in a simple graph on an arbitrary vertex set
- Every walk contains a path between the same endpoints, of no greater length
- The path metric of a connected simple graph
- Metric space: $d(x,y) = 0$ iff $x = y$, symmetry, and the triangle inequality; pseudometric and ultrametric
Used by
- A bijection of vertex sets is an isometry for the path metrics if and only if it is a graph isomorphism Lemma
- On a finite vertex set the graph notions agree, and on connected graphs the two path distances agree Lemma
- The vertex set of a connected simple graph with its path metric is a (1,1)-quasi-geodesic space Proposition
- The word metric is a left-invariant metric and coincides with the path metric of the Cayley graph Theorem
Dependency tree · two levels
16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- C. Loh, Geometric Group Theory: An Introduction (2015 course version), 264 pp. (standard reference, not scraped)
- C. Drutu and M. Kapovich, Geometric Group Theory (with an appendix by B. Nica), 837 pp. (standard reference, not scraped)