On equicut graphs
The size sz(Γ) of an ℓ1-graph Γ = (V, E) is the minimum of nf/tf over all the possible ℓ1-embeddings f into nf -dimensional hypercube with scale tf. The sum of distances between all the pairs of vertices of Γ is at most sz(Γ)⌈v/2⌉⌊v/2⌋ (v = |V |). The latter is an equality if and only if Γ is equicu...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
2011
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/92187 http://hdl.handle.net/10220/6868 http://www.site.uottawa.ca/~ivan/mvl.html |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | The size sz(Γ) of an ℓ1-graph Γ = (V, E) is the minimum of nf/tf over all the possible ℓ1-embeddings f into nf -dimensional hypercube with scale tf. The sum of distances between all the pairs of vertices of Γ is at most sz(Γ)⌈v/2⌉⌊v/2⌋ (v = |V |). The latter is an equality if and only if Γ is equicut graph, that is, Γ admits an ℓ1-embedding f that for any 1 ≤ i ≤ nf satisfies Σx∈X f(x)i ∈ {⌈v/2⌉, ⌊v/2⌋} for any x ∈ V . Basic properties of equicut graphs are investigated. A construction of equicut graphs from ℓ1-graphs via a natural doubling construction is given. It generalizes several well-known constructions of polytopes and distance-regular graphs. Finally, large families of examples, mostly related to polytopes and distance-regular graphs, are presented. |
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