On the 3-Local Profiles of Graphs

Authors


  • Contract grant sponsor: NSF; Contract grant number: DMS-1128155; Contract grant sponsor: Israel Science Foundation; Contract grant sponsor: USA-Israel BSF grant; Contract grant sponsor: NSF; Contract grant number: DMS-1101185; Contract grant sponsor: AFOSR MURI; Contract grant number: FA9550-10-1-0569; Contract grant sponsor: USA-Israel BSF grant.

Abstract

For a graph G, let math formula be the probability that three distinct random vertices span exactly i edges. We call math formula the 3-local profile of G. We investigate the set math formula of all vectors math formula that are arbitrarily close to the 3-local profiles of arbitrarily large graphs. We give a full description of the projection of math formula to the math formula plane. The upper envelope of this planar domain is obtained from cliques on a fraction of the vertex set and complements of such graphs. The lower envelope is Goodman's inequality math formula. We also give a full description of the triangle-free case, i.e. the intersection of math formula with the hyperplane math formula. This planar domain is characterized by an SDP constraint that is derived from Razborov's flag algebra theory.

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