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Tutte polynomials of two self-similar network models. (English) Zbl 1412.82011
Summary: 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.
MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
05C05 Trees
05C82 Small world graphs, complex networks (graph-theoretic aspects)
05C31 Graph polynomials
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[1] Tutte, WT, A contribution to the theory of chromatic polynomials, Can. J. Math., 6, 80-91, (1954) · Zbl 0055.17101
[2] Potts, RB, Some generalized order-disorder transformations, Math. Proc. Camb. Philos. Soc., 48, 106-109, (1952) · Zbl 0048.45601
[3] Wu, FY, The Potts model, Rev. Mod. Phys., 54, 235-268, (1982)
[4] Bak, P.; Tang, C.; Wiesenfeld, K., Self-organized criticality: an explanation of 1/f noise, Phys. Rev. Lett., 59, 381-384, (1987)
[5] Dhar, D., Self-organized critical state of sandpile automaton models, Phys. Rev. Lett., 64, 1613-1616, (1990) · Zbl 0943.82553
[6] Jin, XA; Zhang, FJ, Zeros of the Jones polynomial for multiple crossing-twisted links, J. Stat. Phys., 140, 1054-1064, (2010) · Zbl 1295.57016
[7] Ellis-Monaghan, JA; Merino, C.; Dehmer, M. (ed.), Graph polynomials and their applications I: the Tutte polynomial, 219-255, (2011), Boston · Zbl 1221.05002
[8] Welsh, DJA; Merino, C., The Potts model and the Tutte polynomial, J. Math. Phys., 41, 1127-1152, (2000) · Zbl 1045.82501
[9] Griffiths, RB; 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)
[10] Hu, B., Problem of universality in phase transitions on hierarchical lattices, Phys. Rev. Lett., 55, 2316-2319, (1985)
[11] Yang, ZR, Family of diamond-type hierarchical lattices, Phys. Rev. B, 38, 728-731, (1988)
[12] Qin, Y.; Yang, ZR, Diamond-type hierarchical lattices for the Potts antiferromagnet, Phys. Rev. B, 43, 8576-8582, (1991)
[13] Silva, L., Criticality and multifractality of the Potts ferromagnetic model on fractal lattices, Phys. Rev. B, 53, 6345-6354, (1996)
[14] Muzy, PT; Salinas, SR, Ferromagnetic Potts model on a hierarchical lattice with random layered interactions, Int. J. Mod. Phys. B, 4, 397-409, (1999)
[15] Bleher, PM; Lyubich, MY, Julia Sets and complex singularities in hierarchical Ising models, Commun. Math. Phys., 141, 453-474, (1991) · Zbl 0737.30019
[16] Qiao, JY, Julia sets and complex singularities in diamondlike hierarchical Potts models, Sci. China Ser. A, 48, 388-412, (2005) · Zbl 1170.82331
[17] Ma, F.; Yao, B., The number of spanning trees of self-similar fractal models, Inf. Process. Lett., 136, 64-69, (2018) · Zbl 06873290
[18] Cvetković, D., Rowlinson, P., Simić, S.: An Introduction to the Theory of Graph Spectra. Cambridge University Press, Cambridge (2010) · Zbl 1211.05002
[19] Chang, SC; 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, 667-699, (2009) · Zbl 1191.82008
[20] 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, 215202, (2012) · Zbl 1244.05121
[21] Chang, SC; Shrock, R., Exact partition functions for the q-state Potts model with a generalized magnetic field on lattice strip graphs, J. Stat. Phys., 161, 915-932, (2015) · Zbl 1329.82021
[22] Alvarez, PD; Canfora, F.; Reyes, SA; Riquelme, S., Potts model on recursive lattices: some new exact results, Eur. Phys. J. B, 85, 99, (2012)
[23] Peng, JH; Xiong, J.; Xu, GA, Tutte polynomial of pseudofractal scale-free web, J. Stat. Phys., 159, 1196-1215, (2015) · Zbl 1323.05070
[24] Liao, YH; Hou, YP; Shen, XL, Tutte polynomial of the apollonian network, J. Stat. Mech., 10, p10043, (2014)
[25] Chen, HL; Deng, HY, Tutte polynomial of scale-free networks, J. Stat. Phys., 163, 714-732, (2016) · Zbl 1342.82022
[26] Gong, HL; Jin, XA, A general method for computing Tutte polynomials of self-similar graphs, Physica A, 483, 117-129, (2017)
[27] Lin, Y.; Wu, B.; Zhang, ZZ; Chen, G., Counting spanning trees in self-similar networks by evaluating determinants, J. Math. Phys., 52, 113303, (2011) · Zbl 1272.05187
[28] Rozenfeld, H.; Havlin, S.; Ben-Avraham, D., Fractal and transfractal recursive scale-free net, New J. Phys., 9, 175, (2007)
[29] Jing, H.; Shu, CL, On the normalized Laplacian, degree-Kirchhoff index and spanning trees of graphs, Bull. Aust. Math. Soc., 91, 353-367, (2015) · Zbl 1326.05082
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