On Motzkin–Straus type results for non-uniform hypergraphs

Recently, some extensions of Motzkin–Straus theorems were proved for non-uniform hypergraphs whose edges contain 1 or r vertices in Gu et al. (J Comb Optim 31:223–238, 2016), Peng et al. (Discret Appl Math 200:170–175, 2016a), where r is a given integer. It would be interesting if similar results hold for other non-uniform hypergraphs. In this paper, we establish some Motzkin–Straus type results for general non-uniform hypergraphs. In particular, we obtain some Motzkin–Straus type results in terms of the Lagrangian of non-uniform hypergraphs when there exist some edges consisting of 2 vertices in the given hypergraphs. The presented results unify some known Motzkin–Straus type results for both uniform and non-uniform hypergraphs and also provide solutions to a class of polynomial optimization problems over the standard simplex in Euclidean space.