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.
