Minimum number of below average triangles in a weighted complete graph

Authors:
Gareth Bendall;FrançOis Margot
Affiliations:
Department of Mathematics, University of Kentucky, 715 POT, Lexington, KY 40506, United States;Department of Mathematics, University of Kentucky, 715 POT, Lexington, KY 40506, United States
Venue:
Discrete Optimization
Year:
2006

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Abstract

Let G be an edge weighted graph with n nodes, and let A(3,G) be the average weight of a triangle in G. We show that the number of triangles with weight at most equal to A(3,G) is at least (n-2) and that this bound is sharp for all n=7. Extensions of this result to cliques of cardinality k3 are also discussed.