Abstract
In this paper, we establish a connection between the local maximizers (global maximizers) of a Motzkin-Straus quadratic program and a specific type of regular multipartite cliques. Our work extends a remarkable connection between the maximum cliques and the global maximizers of the Motzkin-Straus quadratic program. This connection and its extensions can be successfully employed in optimization to provide heuristics for the maximal cliques in graphs. We provide two counterexamples to the results from previous work about the global and local maximizers of the Motzkin-Straus quadratic program. We then amend the previous theorems by introducing a new structure. We also answer two questions raised by Pelillo and Jagota about the maxima of the Motzkin-Straus quadratic program.
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Notes
While the Lagrangian is typically used as a function, in extremal graph theory, the Lagrangian for a graph is the maximum value of a polynomial function over the standard simplex. This definition was first used by Motzkin and Straus in the classical paper: Maxima for graphs and a new proof of a theorem of Turán.
In the more standard terminology (and that adopted here), the (n, k)-Turán graph, denoted T(n, k), is the extremal graph on n graph vertices that contains no \((k+1)\)-clique for \(1 \le k \le n\) (Diestel 1997 [6], p. 149; Bollobás 1998, p. 108 [2], and Gross and Yellen 2006, p. 476 [8]). Unfortunately, some authors, including Skiena (1990, pp. 143-144 [24]), Aigner (1995) [1], and Pemmaraju and Skiena (2003, pp. 247-248 [20]), use the convention that the (n, k)-Turán graph contains no k-clique (instead of \((k+1)\)-clique), meaning that the T(n, k)-Turán graph of these authors is the \((n,k-1)\)-Turán graph as defined above.
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Acknowledgements
This research is partially supported by Chinese Universities Scientific Fund (No.N180504008). We thank both reviewers for reading the manuscript carefully, checking all the details and giving insightful comments to help improve the manuscript. We are thankful to a reviewer for pointing out a much simpler approach in the proof of Theorem 5. The proof of Theorem 5 in the current version is based on this reviewer’s suggestion.
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Tang, Q., Zhang, X., Zhao, C. et al. On the maxima of motzkin-straus programs and cliques of graphs. J Glob Optim 84, 989–1003 (2022). https://doi.org/10.1007/s10898-022-01187-3
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DOI: https://doi.org/10.1007/s10898-022-01187-3