Abstract
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.
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References
Bomze IM (1997) Evolution towards the maximum clique. J Glob Optim 10:143–164
Budinich M (2003) Exact bounds on the order of the maximum clique of a graph. Discret Appl Math 127:535–543
Busygin S (2006) A new trust region technique for the maximum weight clique problem. Discret Appl Math 154:2080–2096
Bulò SR, Pelillo M (2009) A generalization of the Motzkin–Straus theorem to hypergraphs. Optim Lett 3:287–295
Frankl P, Füredi Z (1989) Extremal problems whose solutions are the blow-ups of the small Witt-designs. J Combin Theory Ser A 52:129–147
Frankl P, Rödl V (1984) Hypergraphs do not jump. Combinatorica 4:149–159
Gibbons LE, Hearn DW, Pardalos PM, Ramana MV (1997) Continuous characterizations of the maximum clique problem. Math Oper Res 22:754–768
Gu R, Li X, Peng Y, Shi Y (2016) Some Motzkin–Straus type results for non-uniform hypergraphs. J Comb Optim 31:223–238
Motzkin TS, Straus EG (1965) Maxima for graphs and a new proof of a theorem of Turán. Can J Math 17:533–540
Pardalos PM, Phillips A (1990) A global optimization approach for solving the maximum clique problem. Int J Comput Math 33:209–216
Pardalos PM, Pelillo M (2007) Dominant sets and pairwise clustering. IEEE Trans Pattern Anal Mach Intell 29:167–172
Peng Y, Peng H, Tang Q, Zhao C (2016a) An extension of Motzkin–Straus theorem to non-uniform hypergraphs and its applications. Discret Appl Math 200:170–175
Peng Y, Tang Q, Zhao C (2015) On Lagrangians of \(r\)-uniform hypergraphs. J Comb Optim 30:812–825
Peng Y, Wu B, Yao Y (2016b) A note on generalized Lagrangians of non-uniform hypergraphs. Order. doi:10.1007/s11083-016-9385-0
Sidorenko AF (1987) Solution of a problem of Bollobás on 4-graphs. Mat Zametki 41:433–455
Sós VT, Straus EG (1982) Extremals of functions on graphs with applications to graphs and hypergraphs. J Combin Theory Ser A 32:246–257
Acknowledgments
Qingsong Tang was Supported by Chinese Universities Fund (No. N140504004) and Yuejian Peng by National Natural Science Foundation of China (No. 11271116). We thank two anonymous referees for helpful comments.
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Tang, Q., Peng, Y., Zhang, X. et al. On Motzkin–Straus type results for non-uniform hypergraphs. J Comb Optim 34, 504–521 (2017). https://doi.org/10.1007/s10878-016-0084-y
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DOI: https://doi.org/10.1007/s10878-016-0084-y