On equicut graphs
Abstract
The size sz(G) of an l1-graph G=(V,E) is the minimum of nf/tf over all its possible l1-embeddings f into nf-dimensional hypercube with scale tf. In terms of v=|V|, the sum of distances between all the pairs of vertices of G is at most sz(G) v2/4 for v even, (resp. sz(G)(v-1)(v+1)/4 for v odd). This bound is reached if and only if G is an equicut graph, that is, G admits an l1-embedding with column sums v/2, v even (resp. (v-1)/2 for v odd). 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. Large families of examples, mostly related to polytopes and distance-regular graphs, are presented.
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