Two-point resistances in the generalized phenylenes. (English) Zbl 1448.78040
Summary: The resistance between two nodes in some electronic networks has been studied extensively. Let \(G_n\) be a generalized phenylene with \(n\) 6-cycles and \(n\) 4-cycles. Using series and parallel rules and the \(\Delta - Y\) transformations we obtain explicit formulae for the resistance distance between any two points of \(G_n\). To the best of our knowledge \(\{G_n\}_{n=1}^\infty\) is a nontrivial family with diameter going to \(\infty\) for which all resistance distances have been explicitly calculated. We also determine the maximal resistance distance and the minimal resistance distance in \(G_n\). The monotonicity and some asymptotic properties of resistance distances in \(G_n\) are given. At last some numerical results are discussed, in which we calculate the Kirchhoff indices of a set of benzenoid hydrocarbons; We compare their Kirchhoff indices with some other distance-based topological indices through their correlations with the chemical properties. The linear model for the Kirchhoff index is better than or as good as the models corresponding to the other distance-based indices.
MSC:
| 78A57 | Electrochemistry |
| 78A55 | Technical applications of optics and electromagnetic theory |
| 78M35 | Asymptotic analysis in optics and electromagnetic theory |
| 05C12 | Distance in graphs |
| 05C09 | Graphical indices (Wiener index, Zagreb index, Randić index, etc.) |
| 05C10 | Planar graphs; geometric and topological aspects of graph theory |
| 05C90 | Applications of graph theory |
Software:
MathematicaReferences:
| [1] | Ashrafi, AR; Ghorbani, M.; Gutman, I.; Furtula, B., Eccentric connectivity index of fullerences, Novel Molecular Structure Descriptors-Theory and Applications II, 183-192 (2010), Kragujevac: University of Kragujevac, Kragujevac |
| [2] | Bapat, RB; Gupta, S., resistance distance in wheels and fans, Indian J. Pure Appl. Math., 41, 1, 1-13 (2010) · Zbl 1203.05041 · doi:10.1007/s13226-010-0004-2 |
| [3] | Barrett, W.; Evans, EJ; Francis, AE, Resistance distance in straight linear 2-trees, Discrete Appl. Math., 258, 13-34 (2019) · Zbl 1409.05072 · doi:10.1016/j.dam.2018.10.043 |
| [4] | Carmona, A.; Encinas, AM; Mitjana, M., Effective resistances for ladder-like chains, Int. J. Quantum Chem., 114, 24, 16-70 (2014) · doi:10.1002/qua.24740 |
| [5] | A.K. Chandra, P.R. Raghavan, W.L. Ruzzo, R. Smolensky, P. Tiwari, in Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing (Washington, Seattle, 1989), pp. 574-586 |
| [6] | Chen, HY; Zhang, FJ, Resistance distance and the normalized Laplacian spectrum, Discrete Appl. Math., 155, 654-661 (2007) · Zbl 1113.05062 · doi:10.1016/j.dam.2006.09.008 |
| [7] | Chen, HY; Zhang, FJ, Resistance distance local rules, J. Math. Chem., 44, 2, 405-417 (2008) · Zbl 1217.05082 · doi:10.1007/s10910-007-9317-8 |
| [8] | Cinkir, Z., Effective resistances and Kirchhoff index of ladder graphs, J. Math. Chem., 54, 955-966 (2016) · Zbl 1356.05032 · doi:10.1007/s10910-016-0597-8 |
| [9] | Doyle, PG; Snell, JL, Random Walks and Electrical Networks (1984), Washington: The Mathematical Association of America, Washington · Zbl 0583.60065 |
| [10] | Georgakopoulos, A., Uniqueness of electrical currents in a network of finite total resistance, J. Lond. Math. Soc., 82, 1, 256-272 (2010) · Zbl 1209.05116 · doi:10.1112/jlms/jdq034 |
| [11] | Georgakopoulos, A.; Wagner, S., Hitting times, cover cost, and the Wiener index of a tree, J. Graph Theory, 84, 311-326 (2017) · Zbl 1359.05113 · doi:10.1002/jgt.22029 |
| [12] | Gervacio, SV, Resistance distance in complete \(n\)-partite graphs, Discrete Appl. Math., 203, 53-61 (2016) · Zbl 1332.05044 · doi:10.1016/j.dam.2015.09.017 |
| [13] | Huang, J.; Li, SC, On the normalised Laplacian spectrum, degree-Kirchhoff index and spanning trees of graphs, Bull. Aust. Math. Soc., 91, 353-367 (2015) · Zbl 1326.05082 · doi:10.1017/S0004972715000027 |
| [14] | Huang, J.; Li, SC; Xie, Z., Further results on the expected hitting time, the cover cost and the related invariants of graphs, Discrete Math., 342, 78-95 (2019) · Zbl 1400.05218 · doi:10.1016/j.disc.2018.09.019 |
| [15] | Huang, QY; Chen, HY; Deng, QY, Resistance distances and the Kirchhoff index in double graphs, J. Appl. Math. Comput., 50, 1-14 (2016) · Zbl 1330.05052 · doi:10.1007/s12190-014-0855-5 |
| [16] | Huang, S.; Zhou, J.; Bu, C., Some results on Kirchhoff index and degree-Kirchhoff index, MATCH Commun. Math. Comput. Chem., 75, 207-222 (2016) · Zbl 1461.05058 |
| [17] | Jafarizadeh, S.; Sufiani, R.; Jafarizadeh, MA, Evaluation of effective resistances in pseudo-distance-regular resistor networks, J. Stat. Phys., 139, 177-199 (2010) · Zbl 1191.82026 · doi:10.1007/s10955-009-9909-8 |
| [18] | Jafarizadeh, MA; Sufiani, R.; Jafarizadeh, S., Recursive calculation of effective resistances in distance-regular networks based on Bose-Mesner algebra and Christoffel-Darboux identity, J. Math. Phys., 50, 023-302 (2009) · Zbl 1202.94230 · doi:10.1063/1.3077145 |
| [19] | Jiang, ZZ; Yan, WG, Some two-point resistances of the Sierpinski gasket network, J. Stat. Phys., 172, 3, 824-832 (2018) · Zbl 1396.05031 · doi:10.1007/s10955-018-2067-0 |
| [20] | Klein, DJ, Resistance-distance sum rules, Croat. Chem. Acta, 75, 633-649 (2002) |
| [21] | Klein, DJ, Centrality measure in graphs, J. Math. Chem., 47, 1209-1223 (2010) · Zbl 1408.05051 · doi:10.1007/s10910-009-9635-0 |
| [22] | Klein, DJ; Ivanciuc, O., Graph cyclicity, excess conductance, and resistance deficit, J. Math. Chem., 30, 271-287 (2002) · Zbl 1008.05082 · doi:10.1023/A:1015119609980 |
| [23] | Klein, DJ; Lukovits, I.; Gutman, I., On the definition of the hyper-Wiener index for cycle-containing structures, J. Chem. Inf. Comput. Sci., 35, 50-52 (1995) · doi:10.1021/ci00023a007 |
| [24] | Klein, DJ; Randić, M., Resistance distance, J. Math. Chem., 12, 81-95 (1993) · doi:10.1007/BF01164627 |
| [25] | Luxburg, UV; Radl, A.; Hein, M.; Lafferty, JD; Williams, CKI; Shawe-Taylor, J.; Zemel, RS; Culotta, A., Getting lost in space: large sample analysis of the resistance distance, Advances in Neural Information Processing Systems, 2622-2630 (2010), Red Hook: Curran Associates, Inc, Red Hook |
| [26] | Lukovits, I.; Nikolic, S.; Trinajstic, N., Resistance distance in regular graphs, Int. J. Quantum Chem., 71, 217-225 (1999) · doi:10.1002/(SICI)1097-461X(1999)71:3<217::AID-QUA1>3.0.CO;2-C |
| [27] | Maxwell, JC, Electricity and Magnetism (1892), Oxford: Clarendon Press, Oxford |
| [28] | Mieghem, PV, Graph Spectra of Complex Networks (2010), Cambridge: Cambridge University Press, Cambridge |
| [29] | Palacios, JL, Closed-form formulae for Kirchhoff index, Int. J. Quantum Chem., 81, 135-140 (2001) · doi:10.1002/1097-461X(2001)81:2<135::AID-QUA4>3.0.CO;2-G |
| [30] | Peng, YJ; Li, SC, On the Kirchhoff index and the number of spanning trees of linear phenylenes, MATCH Commun. Math. Comput. Chem., 77, 3, 765-780 (2017) · Zbl 1466.92280 |
| [31] | Ramane, HS; Yalnaik, AS, Status connectivity indices of graphs and its applications to the boiling point of benzenoid hydrocarbons, J. Appl. Math. Comput., 55, 1-2, 609-627 (2017) · Zbl 1373.05054 · doi:10.1007/s12190-016-1052-5 |
| [32] | Stevenson, W., Elements of Power System Analysis (1975), New York: McGraw Hill, New York |
| [33] | Shangguan, YM; Chen, HY, Two-point resistances in a family of self-similar \((x, y)\)-flower networks, Phys. A, 523, 382391 (2019) · Zbl 07563382 · doi:10.1016/j.physa.2019.02.008 |
| [34] | Sharpe, GE, Theorem on resistive networks, Electron. Lett., 3, 444-445 (1967) · doi:10.1049/el:19670351 |
| [35] | Shi, LY; Chen, HY, Resistance distances in the linear polyomino chain, J. Appl. Math. Comput., 57, 147-160 (2018) · Zbl 1393.05183 · doi:10.1007/s12190-017-1099-y |
| [36] | Sun, L.; Wang, W.; Zhou, J.; Bu, C., Some results on resistance distances and resistance matrices, Linear Multilinear A, 63, 523-533 (2015) · Zbl 1306.05046 · doi:10.1080/03081087.2013.877011 |
| [37] | Tan, ZZ, Resistance Network Model (2011), Xiaan: Xidian University Press, Xiaan |
| [38] | Tetali, P., Random walks and effective resistance of networks, J. Theor. Probab., 4, 101-109 (1991) · Zbl 0722.60070 · doi:10.1007/BF01046996 |
| [39] | Vaskouski, M.; Zadorozhnyuk, A., Resistance distances in Cayley graphs on symmetric groups, Discrete Appl. Math., 227, 121-135 (2017) · Zbl 1365.05131 · doi:10.1016/j.dam.2017.04.044 |
| [40] | Vukičevič, D.; Graovac, A., A note on the comparison of the first and second normalized Zagreb eccentricity indices, Acta Chim. Slov., 57, 524-528 (2010) |
| [41] | Wiener, H., Structural determination of paraffin boiling points, J. Am. Chem. Soc., 69, 17-20 (1947) · doi:10.1021/ja01193a005 |
| [42] | Wolfram Research, Inc., Mathematica, Version 9.0 (Wolfram Research Inc., Champaign, 2012) |
| [43] | Yang, YJ; Klein, DJ, A recursion formula for resistance distances and its applications, Discrete Appl. Math., 161, 2702-2715 (2013) · Zbl 1285.05050 · doi:10.1016/j.dam.2012.07.015 |
| [44] | Yang, YJ; Klein, DJ, Resistance distance-based graph invariants of subdivisions and triangulations of graphs, Discrete Appl. Math., 181, 260-274 (2015) · Zbl 1304.05040 · doi:10.1016/j.dam.2014.08.039 |
| [45] | Yang, YJ; Klein, DJ, Two-point resistances and random walks on stellated regular graphs, J. Phys. A, 52, 7, 075201 (2019) · Zbl 1505.05117 · doi:10.1088/1751-8121/aaf8e7 |
| [46] | Zhang, HP; Yang, YJ, Resistance distance and Kirchhoff index in circulant graphs, Int. J. Quantum Chem., 107, 330-339 (2007) · doi:10.1002/qua.21068 |
| [47] | Zhou, J.; Sun, L.; Bu, C., Resistance characterizations of equiarboreal graphs, Discrete Math., 340, 2864-2870 (2017) · Zbl 1370.05087 · doi:10.1016/j.disc.2017.07.029 |
| [48] | Zhou, J.; Wang, Z.; Bu, C., On the resistance matrix of a graph, Electron. J. Combin., 23, 1-41 (2016) · Zbl 1333.05191 |
| [49] | Zhu, ZX; Liu, JB, The normalized Laplacian, degree-Kirchhoff index and the spanning tree numbers of generalized phenylenes, Discrete Appl. Math., 254, 256-267 (2019) · Zbl 1404.05124 · doi:10.1016/j.dam.2018.06.026 |
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