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Resistance distance and Kirchhoff index of two kinds of double join operations on graphs

  • Weizhong Wang and Tingyan Ma
Published/Copyright: October 17, 2024

Abstract

Let G be a connected graph. The resistance distance between any two vertices of G is defined to be the network effective resistance between them if each edge of G is replaced by a unit resistor. The Kirchhoff index of G is the sum of resistance distances between all pairs of vertices of G. In this paper, we determine the resistance distance and Kirchhoff index of the subdivision double join GS ∨ {G1, G2} and R-graph double join GR ∨ {G1, G2} for a regular graph G and two arbitrary graphs G1, G2, respectively.


Originally published in Diskretnaya Matematika (2024) 36, №3, 29–49 (in Russian).


Funding statement: This research was supported by the National Natural Science Foundation of China (Nos. 11561042, 11961040) and the Natural Science Foundation of Gansu Province (No. 20JR5RA418).

References

[1] Bonchev D., Balaban A. T., Liu X., Klein D. J., “Molecular cyclicity and centricity of polycyclic graphs. I. Cyclicity based on resistance distances or reciprocal distances”, Int. J. Quantum Chem., 50:1 (1994), 1–20.10.1002/qua.560500102Search in Google Scholar

[2] Bapat R. B., Graphs and Matrices, Springer, 171 pp.Search in Google Scholar

[3] Bapat R. B., Gupta S., “Resistance distance in wheels and fans”, Indian J. Pure Appl. Math., 41:1 (2010), 1–13.10.1007/s13226-010-0004-2Search in Google Scholar

[4] Bapat R. B., Gutman I., Xiao W., “A simple method for computing resistance distance”, Z. Naturforsch. A, 58:9-10 (2003), 494–498.10.1515/zna-2003-9-1003Search in Google Scholar

[5] Bu C. J., Sun L. Z., Zhou J., Wei Y. M., “A note on block representations of the group inverse of Laplacian matrices”, Electron. J. Linear Algebra, 23 (2012), 866–876.10.13001/1081-3810.1562Search in Google Scholar

[6] Bu C. J., Yan B., Zhou X. Q., Zhou J., “Resistance distance in subdivision-vertex join and subdivision-edge join of graphs”, Linear Algebra Appl., 458 (2014), 454–462.10.1016/j.laa.2014.06.018Search in Google Scholar

[7] Cardoso D. M., Díaz R. C., Rojo O., “Distance matrices on the H-join of graphs: A general result and applications”, Linear Algebra Appl., 559 (2018), 34–53.10.1016/j.laa.2018.08.024Search in Google Scholar

[8] Cardoso D. M., de Freitas M. A. A., Martins E. A., Robbiano M., “Spectra of graphs obtained by a generalization of the join graph operation”, Discrete Math., 313:5 (2013), 733–741.10.1016/j.disc.2012.10.016Search in Google Scholar

[9] Cardoso D. M., Martins E. A., Robbiano M., Rojo O., “Eigenvalues of a H-generalized join graph operation constrained by vertex subsets”, Linear Algebra Appl., 438:8 (2013), 3278–3290.10.1016/j.laa.2012.12.004Search in Google Scholar

[10] Chen H., “Random walks and the effective resistance sum rules”, Discrete Appl. Math., 158:15 (2010), 1691–1700.10.1016/j.dam.2010.05.020Search in Google Scholar

[11] Das A., Panigrahi P., “Normalized Laplacian spectrum of some subdivision-joins and R-joins of two regular graphs”, AKCE Int. J. Graphs Comb., 15:3 (2018), 261–270.10.1016/j.akcej.2017.10.006Search in Google Scholar

[12] Gutman I., Mohar B., “The quasi-Wiener and the Kirchhoff indices coincide”, J. Chem. Inf. Comput. Sci., 36:5 (1996), 982–985.10.1021/ci960007tSearch in Google Scholar

[13] Indulal G., “Spectrum of two new joins of graphs and infinite families of integral graphs”, Kragujevac J. Math., 36:1 (2012), 133–139.Search in Google Scholar

[14] Kirkland S. J., Neumann M., “The M-matrix group generalized inverse problem for weighted trees”, SIAM J. Matrix Anal. Appl., 19:1 (1998), 226–234.10.1137/S0895479896304927Search in Google Scholar

[15] Klein D. J., Randić M., “Resistance distance”, J. Math. Chem., 12:1 (1993), 81–95.10.1007/BF01164627Search in Google Scholar

[16] Liu Q., Liu J. B., Cao J. D., “The Laplacian polynomial and Kirchhoff index of graphs based on R-graphs”, Neurocomputing, 177 (2016), 441–446.10.1016/j.neucom.2015.11.060Search in Google Scholar

[17] Liu Q., Wang W. Z., “Resistance distance and Kirchhoff index in subdivision-vertex and subdivision-edge neighbourhood coronae”, Ars Comb., 130 (2017), 29–41.Search in Google Scholar

[18] Liu X. G., Zhou J., Bu C. J., “Resistance distance and Kirchhoff index of R-vertex join and R-edge join of two graphs”, Discrete Appl. Math., 187 (2015), 130–139.10.1016/j.dam.2015.02.021Search in Google Scholar

[19] Liu X. G., Zhang Z. H., “Spectra of subdivision-vertex join and subdivision-edge join of two graphs”, Bull. Malays. Math. Sci. Soc., 42 (2019), 15–31.10.1007/s40840-017-0466-zSearch in Google Scholar

[20] Sun L. Z., Wang W. Z., Zhou J., Bu C. J., “Some results on resistance distances and resistance matrices”, Linear Multilinear Algebra, 63:3 (2015), 523–533.10.1080/03081087.2013.877011Search in Google Scholar

[21] Tian G. X., He J. X., Cui S. Y., “On the Laplacian spectra of some double join operations of graphs”, Bull. Malays. Math. Sci. Soc., 42 (2019), 1555–1566.10.1007/s40840-017-0566-9Search in Google Scholar

[22] Wang W. Z., Yang D., Luo Y. F., “The Laplacian polynomial and Kirchhoff index of graphs derived from regular graphs”, Discrete Appl. Math., 161:18 (2013), 3063–3071.10.1016/j.dam.2013.06.010Search in Google Scholar

[23] Yang Y. J., Klein D. J., “Resistance distance-based graph invariants of subdivisions and triangulations of graphs”, Discrete Appl. Math., 181 (2015), 260–274.10.1016/j.dam.2014.08.039Search in Google Scholar

[24] Yang Y. J., “The Kirchhoff index of subdivisions of graphs”, Discrete Appl. Math., 171 (2014), 153–157.10.1016/j.dam.2014.02.015Search in Google Scholar

[25] Yang Y. J., Zhang H. P., “Some rules on resistance distance with applications”, J. Phys. A: Math. Theor., 41:44 (2008), 445203.10.1088/1751-8113/41/44/445203Search in Google Scholar

[26] Zhu H. Y., Klein D. J., Lukovits I., “Extensions of the Wiener number”, J. Chem. Inf. Comput. Sci., 36:3 (1996), 420–428.10.1021/ci950116sSearch in Google Scholar

[27] Zhang F. Z., The Schur Complement and Its Applications, Numer. Meth. Algorithms, 4, Springer, 2005, xvi + 295 pp.10.1007/b105056Search in Google Scholar

[28] Zhang H. P., Yang Y. J., Li C. W., “Kirchhoff index of composite graphs”, Discrete Appl. Math., 157:13 (2009), 2918–2927.10.1016/j.dam.2009.03.007Search in Google Scholar

[29] Zhang H. P., Yang Y. J., “Resistance distance and Kirchhoff index in circulant graphs”, Int. J. Quantum Chem., 107:2 (2007), 330–339.10.1002/qua.21068Search in Google Scholar

Received: 2022-04-04
Published Online: 2024-10-17
Published in Print: 2024-10-28

© 2024 Walter de Gruyter GmbH, Berlin/Boston

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