Abstract
This work introduces new bounds on the clique number of graphs derived from a result due to Sós and Straus, which generalizes the Motzkin-Straus Theorem to a specific class of hypergraphs. In particular, we generalize and improve the spectral bounds introduced by Wilf in 1967 and 1986 establishing an interesting link between the clique number and the emerging spectral hypergraph theory field. In order to compute the bounds we face the problem of extracting the leading H-eigenpair of supersymmetric tensors, which is still uncovered in the literature. To this end, we provide two approaches to serve the purpose. Finally, we present some preliminary experimental results.
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References
Hastad, J.: Clique is hard to approximate within n 1 − ε. Ann. Symp. Found. Comput. Sci. 37, 627–636 (1996)
Lovás, L.: On the Shannon capacity of a graph. IEEE Trans. Inform. Theory IT-25, 1–7 (1979)
Schrijver, A.: A comparison of the Delsarte and Lovász bounds. IEEE Trans. Inform. Theory 25, 425–429 (1979)
Wilf, H.S.: The eigenvalues of a graph and its chromatic number. J. London Math. Soc. 42, 330–332 (1967)
Motzkin, T.S., Straus, E.G.: Maxima for graphs and a new proof of a theorem of Turán. Canad. J. Math. 17, 533–540 (1965)
Wilf, H.S.: Spectral bounds for the clique and independence numbers of graphs. J. Combin. Theory Series B 40, 113–117 (1986)
Budinich, M.: Exact bounds on the order of the maximum clique of a graph. Discr. Appl. Math. 127, 535–543 (2003)
Lu, M., Liu, H., Tian, F.: Laplacian spectral bounds for clique and independence numbers of graphs. J. Combin. Theory Series B 97(5), 726–732 (2007)
Sós, V., Straus, E.G.: Extremal of functions on graphs with applications to graphs and hypergraphs. J. Combin. Theory Series B 63, 189–207 (1982)
Lim, L.H.: Singular values and eigenvalues of tensors: a variational approach. In: IEEE Int. Workshop on Comput. Adv. in Multi-Sensor Adapt. Processing, vol. 1, pp. 129–132 (2005)
Qi, L.: Eigenvalues of a real supersymmetric tensor. J. of Symb. Comp. 40, 1302–1324 (2005)
Baratchart, L., Berthod, M., Pottier, L.: Optimization of positive generalized polynomials under ℓp constraints. J. of Convex Analysis 5(2), 353–379 (1998)
Berthod, M.: Definition of a consistent labeling as a global extremum. In: Int. Conf. Patt. Recogn., pp. 339–341 (1982)
Baum, L.E., Eagon, J.A.: An inequality with applications to statistical estimation for probabilistic functions of Markov processes and to a model for ecology. Bull. Amer. Math. Soc. 73, 360–363 (1967)
Rota Bulò, S.: A game-theoretic framework for similarity-based data clustering. PhD Thesis. Work in progress (2009)
Amin, A.T., Hakimi, S.L.: Upper bounds of the order of a clique of a graph. SIAM J. on Appl. Math. 22(4), 569–573 (1972)
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Rota Bulò, S., Pelillo, M. (2009). New Bounds on the Clique Number of Graphs Based on Spectral Hypergraph Theory. In: Stützle, T. (eds) Learning and Intelligent Optimization. LION 2009. Lecture Notes in Computer Science, vol 5851. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-11169-3_4
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DOI: https://doi.org/10.1007/978-3-642-11169-3_4
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