Abstract
The Tutte polynomial T(G; x, y) of a graph G, or equivalently the q-state Potts model partition function, is a two-variable polynomial graph invariant of considerable importance in combinatorics and statistical physics. Graph operations have been extensively applied to model complex networks recently. In this paper, we study the Tutte polynomials of the diamond hierarchical lattices and a class of self-similar fractal models which can be constructed through graph operations. Firstly, we find out the behavior of the Tutte polynomial under k-inflation and k-subdivision which are two graph operations. Secondly, we compute and gain the Tutte polynomials of this two self-similar fractal models by using their structure characteristic. Moreover, as an application of the obtained results, some evaluations of their Tutte polynomials are derived, such as the number of spanning trees and the number of spanning forests.
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References
Tutte, W.T.: A contribution to the theory of chromatic polynomials. Can. J. Math. 6, 80–91 (1954)
Potts, R.B.: Some generalized order-disorder transformations. Math. Proc. Camb. Philos. Soc. 48(1), 106–109 (1952)
Wu, F.Y.: The Potts model. Rev. Mod. Phys. 54(1), 235–268 (1982)
Bak, P., Tang, C., Wiesenfeld, K.: Self-organized criticality: an explanation of 1/f noise. Phys. Rev. Lett. 59(4), 381–384 (1987)
Dhar, D.: Self-organized critical state of sandpile automaton models. Phys. Rev. Lett. 64, 1613–1616 (1990)
Jin, X.A., Zhang, F.J.: Zeros of the Jones polynomial for multiple crossing-twisted links. J. Stat. Phys. 140(6), 1054–1064 (2010)
Ellis-Monaghan, J.A., Merino, C.: Graph polynomials and their applications I: the Tutte polynomial. In: Dehmer, M. (ed.) Structural Analysis of Complex Networks, pp. 219–255. Birkhüser, Boston (2011)
Welsh, D.J.A., Merino, C.: The Potts model and the Tutte polynomial. J. Math. Phys. 41, 1127–1152 (2000)
Griffiths, R.B., Kaufman, M.: First-order transitions in defect structures at a second-order critical point for the Potts model on hierarchical lattices. Phys. Rev. B 26, 5022–5032 (1982)
Hu, B.: Problem of universality in phase transitions on hierarchical lattices. Phys. Rev. Lett. 55, 2316–2319 (1985)
Yang, Z.R.: Family of diamond-type hierarchical lattices. Phys. Rev. B 38, 728–731 (1988)
Qin, Y., Yang, Z.R.: Diamond-type hierarchical lattices for the Potts antiferromagnet. Phys. Rev. B 43, 8576–8582 (1991)
de Silva, L.: Criticality and multifractality of the Potts ferromagnetic model on fractal lattices. Phys. Rev. B 53, 6345–6354 (1996)
Muzy, P.T., Salinas, S.R.: Ferromagnetic Potts model on a hierarchical lattice with random layered interactions. Int. J. Mod. Phys. B 4, 397–409 (1999)
Bleher, P.M., Lyubich, M.Y.: Julia Sets and complex singularities in hierarchical Ising models. Commun. Math. Phys. 141, 453–474 (1991)
Qiao, J.Y.: Julia sets and complex singularities in diamondlike hierarchical Potts models. Sci. China Ser. A 48, 388–412 (2005)
Ma, F., Yao, B.: The number of spanning trees of self-similar fractal models. Inf. Process. Lett. 136, 64–69 (2018)
Cvetković, D., Rowlinson, P., Simić, S.: An Introduction to the Theory of Graph Spectra. Cambridge University Press, Cambridge (2010)
Chang, S.C., Shrock, R.: Structure of the partition function and transfer matrices for the Potts model in a magnetic field on lattice strips. J. Stat. Phys. 137(4), 667–699 (2009)
Shrock, R., Xu, Y.: The structure of chromatic polynomials of planar triangulation graphs and implications for chromatic zeros and asymptotic limiting quantities. J. Phys. A 45(21), 215202 (2012)
Chang, S.C., Shrock, R.: Exact partition functions for the q-state Potts model with a generalized magnetic field on lattice strip graphs. J. Stat. Phys. 161(4), 915–932 (2015)
Alvarez, P.D., Canfora, F., Reyes, S.A., Riquelme, S.: Potts model on recursive lattices: some new exact results. Eur. Phys. J. B 85(3), 99 (2012)
Peng, J.H., Xiong, J., Xu, G.A.: Tutte polynomial of pseudofractal scale-free web. J. Stat. Phys. 159(5), 1196–1215 (2015)
Liao, Y.H., Hou, Y.P., Shen, X.L.: Tutte polynomial of the apollonian network. J. Stat. Mech. 10, P10043 (2014)
Chen, H.L., Deng, H.Y.: Tutte polynomial of scale-free networks. J. Stat. Phys. 163(4), 714–732 (2016)
Gong, H.L., Jin, X.A.: A general method for computing Tutte polynomials of self-similar graphs. Physica A 483, 117–129 (2017)
Lin, Y., Wu, B., Zhang, Z.Z., Chen, G.: Counting spanning trees in self-similar networks by evaluating determinants. J. Math. Phys. 52, 113303 (2011)
Rozenfeld, H., Havlin, S., Ben-Avraham, D.: Fractal and transfractal recursive scale-free net. New J. Phys. 9, 175 (2007)
Jing, H., Shu, C.L.: On the normalized Laplacian, degree-Kirchhoff index and spanning trees of graphs. Bull. Aust. Math. Soc. 91, 353–367 (2015)
Acknowledgements
The authors thank the anonymous referees for their valuable comments and helpful suggestions, which have considerably improved the presentation of this paper. This work was supported by the National Natural Science Foundation of China (No. 11571101), the Scientific Research Fund of Hunan Provincial Education Department (No. 16C0872) and the Hunan Provincial Natural Science Foundation of China (No. 2018JJ3255). Yunhua Liao and M. A. Aziz-Alaoui were supported by Normandie region France and the XTerm ERDF project (European Regional Development Fund) on Complex Networks and Applications.
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Liao, Y., Xie, X., Hou, Y. et al. Tutte Polynomials of Two Self-similar Network Models. J Stat Phys 174, 893–905 (2019). https://doi.org/10.1007/s10955-018-2204-9
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DOI: https://doi.org/10.1007/s10955-018-2204-9