About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 794781, 7 pages
http://dx.doi.org/10.1155/2014/794781
Research Article

Some Bounds for the Kirchhoff Index of Graphs

School of Mathematics and Information Science, Yantai University, Yantai 264005, China

Received 7 February 2014; Accepted 24 June 2014; Published 10 July 2014

Academic Editor: Massimo Furi

Copyright © 2014 Yujun Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. D. J. Klein and M. Randić, “Resistance distance,” Journal of Mathematical Chemistry, vol. 12, no. 1–4, pp. 81–95, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  2. B. Zhou and N. Trinajstić, “Mathematical properties of molecular descriptors based on distances,” Croatica Chemica Acta, vol. 83, pp. 227–242, 2010.
  3. H. Wiener, “Structural determination of paraffin boiling points,” Journal of the American Chemical Society, vol. 69, no. 1, pp. 17–20, 1947. View at Publisher · View at Google Scholar · View at Scopus
  4. D. Babić, D. J. Klein, I. Lukovits, S. Nikolić, and N. Trinajstić, “Resistance-distance matrix: a computational algorithm and its application,” International Journal of Quantum Chemistry, vol. 90, no. 1, pp. 166–176, 2002. View at Publisher · View at Google Scholar
  5. M. Bianchi, A. Cornaro, J. L. Palacios, and A. Torriero, “Bounds for the Kirchhoff index via majorization techniques,” Journal of Mathematical Chemistry, vol. 51, no. 2, pp. 569–587, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  6. A. Cornaro and G. P. Clemente, “A new lower bound for the Kirchhoff index using a numerical procedure based on majorization techniques,” Electronic Notes in Discrete Mathematics, vol. 41, pp. 383–390, 2013. View at Publisher · View at Google Scholar · View at Scopus
  7. K. C. Das, “On the Kirchhoff index of graphs,” Zeitschrift für Naturforschung, vol. 68, pp. 531–538, 2013. View at Publisher · View at Google Scholar
  8. K. C. Das, A. D. Güngörr, and A. S. Çevik, “On Kirchhoff index and resistance-distance energy of a graph,” MATCH: Communications in Mathematical and in Computer Chemistry, vol. 67, no. 2, pp. 541–556, 2012. View at MathSciNet
  9. K. C. Das, K. Xu, and I. Gutman, “Comparison between Kirchhoff index and the Laplacian-energy-like invariant,” Linear Algebra and its Applications, vol. 436, no. 9, pp. 3661–3671, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. I. Gutman, D. Vidović, and B. Furtula, “Chemical applications of the Laplacian spectrum. VII. Studies of the Wiener and Kirchhoff indices,” Indian Journal of Chemistry A: Inorganic, Physical, Theoretical and Analytical Chemistry, vol. 42, no. 6, pp. 1272–1278, 2003. View at Scopus
  11. R. Li, “Lower bounds for the Kirchhoff index,” MATCH: Communications in Mathematical and in Computer Chemistry, vol. 70, no. 1, pp. 163–174, 2013. View at MathSciNet
  12. I. Lukovits, S. Nikolić, and N. Trinajstić, “Resistance distance in regular graphs,” International Journal of Quantum Chemistry, vol. 71, no. 3, pp. 217–225, 1999. View at Publisher · View at Google Scholar · View at Scopus
  13. J. L. Palacios, “Closed-form formulas for Kirchhoff index,” International Journal of Quantum Chemistry, vol. 81, pp. 135–140, 2001.
  14. J. L. Palacios, “Foster's formulas via probability and the Kirchhoff index,” Methodology and Computing in Applied Probability, vol. 6, no. 4, pp. 381–387, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. J. L. Palacios and J. M. Renom, “Bounds for the Kirchhoff index of regular graphs via the spectra of their random walks,” International Journal of Quantum Chemistry, vol. 110, no. 9, pp. 1637–1641, 2010. View at Publisher · View at Google Scholar · View at Scopus
  16. J. L. Palacios and J. M. Renom, “Broder and Karlin's formula for hitting times and the Kirchhoff Index,” International Journal of Quantum Chemistry, vol. 111, no. 1, pp. 35–39, 2011. View at Publisher · View at Google Scholar · View at Scopus
  17. B. Zhou, “On sum of powers of the Laplacian eigenvalues of graphs,” Linear Algebra and its Applications, vol. 429, no. 8-9, pp. 2239–2246, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  18. B. Zhou and N. Trinajstić, “A note on Kirchhoff index,” Chemical Physics Letters, vol. 445, pp. 120–123, 2008.
  19. B. Zhou and N. Trinajstić, “The kirchhoff index and the matching number,” International Journal of Quantum Chemistry, vol. 109, no. 13, pp. 2978–2981, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. B. Zhou and N. Trinajstić, “On resistance-distance and Kirchhoff index,” Journal of Mathematical Chemistry, vol. 46, no. 1, pp. 283–289, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  21. L. Feng, G. Yu, K. Xu, and Z. Jiang, “A note on the Kirchhoff index of bicyclic graphs,” Ars Combinatoria, vol. 114, pp. 33–40, 2014.
  22. Q. Guo, H. Deng, and D. Chen, “The extremal Kirchhoff index of a class of unicyclic graphs,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 61, pp. 713–722, 2009.
  23. H. Wang, H. Hua, and D. Wang, “Cacti with minimum, second-minimum, and third-minimum Kirchhoff indices,” Mathematical Communications, vol. 15, no. 2, pp. 347–358, 2010. View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  24. Y. Yang, “Bounds for the Kirchhoff index of bipartite graphs,” Journal of Applied Mathematics, vol. 2012, Article ID 195242, 9 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  25. W. Zhang and H. Deng, “The second maximal and minimal Kirchhoff indices of unicyclic graphs,” MATCH: Communications in Mathematical and in Computer Chemistry, vol. 61, no. 3, pp. 683–695, 2009. View at MathSciNet · View at Scopus
  26. H. Zhang, X. Jiang, and Y. Yang, “Bicyclic graphs with extremal Kirchhoff index,” MATCH: Communications in Mathematical and in Computer Chemistry, vol. 61, no. 3, pp. 697–712, 2009. View at MathSciNet · View at Scopus
  27. H. Zhang and Y. Yang, “Resistance distance and kirchhoff index in circulant graphs,” International Journal of Quantum Chemistry, vol. 107, no. 2, pp. 330–339, 2007. View at Publisher · View at Google Scholar · View at Scopus
  28. Q. Deng and H. Chen, “On the Kirchhoff index of the complement of a bipartite graph,” Linear Algebra and Its Applications, vol. 439, no. 1, pp. 167–173, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. Q. Deng and H. Chen, “On extremal bipartite unicyclic graphs,” Linear Algebra and Its Applications, vol. 444, pp. 89–99, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  30. X. Gao, Y. Luo, and W. Liu, “Kirchhoff index in line, subdivision and total graphs of a regular graph,” Discrete Applied Mathematics, vol. 160, no. 4-5, pp. 560–565, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. J. Liu, J. Cao, X. Pan, and A. Elaiw, “The Kirchhoff index of hypercubes and related complex networks,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 543189, 7 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  32. J. Liu, X. Pan, Y. Wang, and J. Cao, “The Kirchhoff index of folded hypercubes and some variant networks,” Mathematical Problems in Engineering, vol. 2014, Article ID 380874, 9 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  33. X. Kuang and W. Yan, “The Kirchhoff indices of some graphs,” Journal of Jimei University, vol. 17, pp. 65–70, 2012.
  34. M. H. Shirdareh-Haghighi, Z. Sepasdar, and A. Nikseresht, “On the Kirchhoff index of graphs and some graph operations,” Proceedings of the Indian Academy of Science. In press.
  35. W. Wang, D. Yang, and Y. Luo, “The Laplacian polynomial and Kirchhoff index of graphs derived from regular graphs,” Discrete Applied Mathematics, vol. 161, no. 18, pp. 3063–3071, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. Z. You, L. You, and W. Hong, “Comment on ‘Kirchhoff index in line, subdivision and total graphs of a regular graph’,” Discrete Applied Mathematics, vol. 161, no. 18, pp. 3100–3103, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. R. Merris, “Laplacian matrices of graphs: a survey,” Linear Algebra and Its Applications, vol. 197-198, pp. 143–176, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  38. H. Y. Zhu, D. J. Klein, and I. Lukovits, “Extensions of the Wiener number,” Journal of Chemical Information and Computer Sciences, vol. 36, no. 3, pp. 420–428, 1996. View at Scopus
  39. I. Gutman and B. Mohar, “The quasi-Wiener and the Kirchhoff indices coincide,” Journal of Chemical Information and Computer Sciences, vol. 36, no. 5, pp. 982–985, 1996. View at Scopus
  40. O. Ivanciuc and D. J. Klein, “Building-block computation of Wiener-type indices for the virtual screening of combinatorial libraries,” Croatica Chemica Acta, vol. 75, no. 2, pp. 577–601, 2002. View at Scopus
  41. R. C. Entringer, D. E. Jackson, and D. A. Snyder, “Distance in graphs,” Czechoslovak Mathematical Journal, vol. 26, no. 2, pp. 283–296, 1976. View at MathSciNet
  42. W. N. Anderson and T. D. Morley, “Eigenvalues of the Laplacian of a graph,” Linear and Multilinear Algebra, vol. 18, no. 2, pp. 141–145, 1985. View at Publisher · View at Google Scholar · View at MathSciNet
  43. R. M. Foster, “The average impedance of an electrical network,” in Contributions to Applied Mechanics, J. W. Edwards, Ed., pp. 333–340, Edwards Brothers, Ann Arbor, Mich, USA, 1949.
  44. R. M. Foster, “An extension of a network theorem,” IRE Transactions on Circuit Theory, vol. 8, pp. 75–76, 1961.
  45. D. Coppersmith, P. Tetali, and P. Winkler, “Collisions among random walks on a graph,” SIAM Journal on Discrete Mathematics, vol. 6, no. 3, pp. 363–374, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  46. P. G. Doyle and J. L. Snell, Random Walks and Electric Networks, vol. 22 of Carus Mathematical Monographs, Mathematical Association of America, Washington, DC, 1984. View at MathSciNet
  47. A. Behmaram, H. Yousefi-Azari, and A. R. Ashrafi, “Wiener polarity index of fullerenes and hexagonal systems,” Applied Mathematics Letters, vol. 25, no. 10, pp. 1510–1513, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  48. K. Menger, “Zur allgemeinen Kurventhoerie,” Fundamenta Mathematicae, vol. 10, pp. 96–115, 1927.