Abstract
Motzkin and Straus established a remarkable connection between the maximum clique and the Lagrangian of a graph in [8]. They showed that if G is a 2-graph in which a largest clique has order l then \({\lambda(G)=\lambda(K^{(2)}_l),}\) where λ(G) denotes the Lagrangian of G. It is interesting to study a generalization of the Motzkin–Straus Theorem to hypergraphs. In this note, we give a Motzkin–Straus type result. We show that if m and l are positive integers satisfying \({{l-1 \choose 3} \le m \le {l-1 \choose 3} + {l-2 \choose 2}}\) and G is a 3-uniform graph with m edges and G contains a \({K_{l-1}^{(3)}}\), a clique of order l−1, then \({\lambda(G) = \lambda(K_{l-1}^{(3)})}\). Furthermore, the upper bound \({{l-1 \choose 3} + {l-2 \choose 2}}\) is the best possible.
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Peng, Y., Zhao, C. A Motzkin–Straus Type Result for 3-Uniform Hypergraphs. Graphs and Combinatorics 29, 681–694 (2013). https://doi.org/10.1007/s00373-012-1135-5
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DOI: https://doi.org/10.1007/s00373-012-1135-5