Resistance distances and the Kirchhoff index in double graphs

详细信息    查看全文

• 作者：Qinying Huang ; Haiyan Chen ; Qingying Deng
• 关键词：Double graph ; Resistance distance ; Kirchhoff index ; 05C35 ; 05C50
• 刊名：Journal of Applied Mathematics and Computing
• 出版年：2016
• 出版时间：February 2016
• 年：2016
• 卷：50
• 期：1-2
• 页码：1-14
• 全文大小：471 KB
• 参考文献：1.Indulal, G., Vijayakumar, A.: On a pair of equienergetic graphs. MATCH Commun. Math. Comput. Chem. 55, 83–90 (2006)MathSciNet
  2.Munarini, E., Cippo, C.P., Scagliola, A., Salvi, N.Z.: Double graphs. Discrete Math. 308, 242–254 (2008)MathSciNet CrossRef
  3.Marino, M.C., Salvi, N.Z.: Generalizing double graphs, Atti dell Accademia Peloritana dei Pericolanti Classe di Scienze Fisiche. Matematiche e Naturali LXXXV, C1A0702002 (2007)
  4.Xin, L.W., Yi, W.F.: The number of spanning trees of double graphs. Kragujevac J. Math. 35(1), 183–190 (2011)MathSciNet
  5.Indulal, G.: On the distance spectra of some graphs. Math. Commun. 13, 123–131 (2008)MathSciNet
  6.Klein, D.J., Randić, M.: Resistance distance. J. Math. Chem. 12, 81–95 (1993)MathSciNet CrossRef
  7.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, 1–20 (1994)CrossRef
  8.Wiener, H.: Structural determination of paraffin boiling points. J. Am. Chem. Soc. 69, 17–20 (1947)CrossRef
  9.Xiao, W., Gutman, I.: Resistance distance and Laplacian spectrum. Theor. Chem. Acc. 110, 284–289 (2003)CrossRef
  10.Yang, Y.J., Klein, D.J.: A recursion formula for resistance distances and its applications. Discrete Appl. Math. 161, 2702–2715 (2013)MathSciNet CrossRef
  11.Klein, D.J., 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)CrossRef
  12.Lukovits, I., Nikolic, S., Trinajstic, N.: Resistance distance in regular graphs. Int. J. Quantum Chem. 71, 217–225 (1999)CrossRef
  13.Zhang, H.P., Yang, Y.J.: Resistance distance and Kirchhoff index in circulant graphs. Int. J. Quantum Chem. 107, 330–339 (2007)CrossRef
  14.Jafarizadeh, M.A., Sufiani, R., Jafarizadeh, S.: Recursive calculation of effective resistances in distance-regular networks based on Bose–Mesner algebra and Christoffel–Darboux identity. J. Math. Phy. 50, 023302 (2009)MathSciNet CrossRef
  15.Palacios, J.L.: Closed-form formulas for Kirchhoff index. Int. J. Quantum Chem. 81, 135–140 (2001)CrossRef
  16.Jafarizadeh, S., Sufiani, R., Jafarizadeh, M.A.: Evaluation of effective resistances in pseudo-distance-regular resistor networks. J. Stat. Phy. 139, 177–199 (2010)MathSciNet CrossRef
  17.Babat, R.B., Gupta, S.: Resistance distance in wheels and fans. India J. Pure Appl. Math. 41, 1–13 (2010)CrossRef
  18.Gao, X., Luo, Y., Liu, W.: Resistance distances and the Kirchhoff index in Cayley graphs. Discrete Appl. Math. 159, 2050–2057 (2011)MathSciNet CrossRef
  19.Fowler, P.W.: Resistance distances in fullerene graphs. Croat. Chem. Acta 75, 401–408 (2002)
  20.Xu, H.: The Laplacian spectrum and Kirchhoff Index of product and lexicographic product of graphs. J. Xiamen Univ. (Nat. Sci.) 42, 552–554 (2003). (in Chinese)
  21.Zhang, H.P., Yang, Y.J., Li, C.W.: Kirchhoff index of composite graphs. Discrete Appl. Math. 107, 2918–2927 (2009)MathSciNet CrossRef
  22.Gao, X., Luo, Y., Liu, W.: Kirchhoff index in line, subdivision and total graphs of a regular graph. Discrete Appl. Math. 160, 560–565 (2012)MathSciNet CrossRef
  23.Babic, D., Klein, D.J., Lukovits, I., Nikolic, S., Trinajstic, N.: Resistance-distance matrix: A computational algorithm and its application. Int. J. Quantum Chem. 90, 166–176 (2002)CrossRef
  24.Gutman, I., Mohar, B.: The quasi-Wiener and the Kirchhoff indices coincide. J. Chem. Inf. Comput. Sci. 36, 982–985 (1996)CrossRef
  25.Dobrynin, A.A.: Wiener index of trees: theory and applications. Acta Applicandae Mathematicae 66, 211–249 (2001)MathSciNet CrossRef
  26.Entringer, R.C., Jackon, D.E., SZékrly, D.A.: Distance in graphs. Czechoslovak Math. J. 26, 283–296 (1976)MathSciNet
  
• 作者单位：Qinying Huang (1)
  Haiyan Chen (1)
  Qingying Deng (1)
  
  1. School of Science, Jimei University, Xiamen, 361021, Fujian, People’s Republic of China
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Computational Mathematics and Numerical Analysis
  Applied Mathematics and Computational Methods of Engineering
  Theory of Computation
  Mathematics of Computing
  
• 出版者：Springer Berlin / Heidelberg
• ISSN：1865-2085

文摘

Let \(G\) be a connected graph, and let \(DG\) denote the double graph of \(G\). In this paper, we first derive closed-form formulas for resistance distances and the Kirchhoff index of \(DG\) in terms of that of \(G\). Then closed-form formulas for general \(k\)-iterated double graphs are also obtained. Finally, as illustration examples, for several special kinds of graphs, such as, the complete graph, the path, the cycle, etc., the explicit formulas for resistance distances and Kirchhoff indices of their \(k\)-iterated double graphs are given respectively. Keywords Double graph Resistance distance Kirchhoff index