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Discrete Dynamics in Nature and Society
Volume 2018, Article ID 3957023, 6 pages
https://doi.org/10.1155/2018/3957023
Research Article

The Harary Index of All Unicyclic Graphs with Given Diameter

1School of Mathematics and Computation Sciences, Anqing Normal University, Anqing 246133, China
2Basic Department, Hefei Preschool Education College, Hefei 230013, China
3School of Mathematics, Southeast University, Nanjing 210096, China

Correspondence should be addressed to Jinde Cao; nc.ude.ues@oacdj

Received 15 June 2018; Revised 23 July 2018; Accepted 7 August 2018; Published 16 September 2018

Academic Editor: Beatrice Di Bella

Copyright © 2018 Bao-Hua Xing et al. 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. J. A. Bondy and U. S. R. Murty, Graph Theory with Application, Macmillan Press, New York, NY, USA, 1976. View at MathSciNet
  2. O. Ivanciuc, T.-S. Balaban, and A. T. Balaban, “Design of topological indices. Part 4. Reciprocal distance matrix, related local vertex invariants and topological indices,” Journal of Mathematical Chemistry, vol. 12, no. 1-4, pp. 309–318, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  3. D. Plavšić, S. Nikolić, N. Trinajstić, and Z. Mihalić, “On the Harary index for the characterization of chemical graphs,” Journal of Mathematical Chemistry, vol. 12, no. 1-4, pp. 235–250, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  4. J. Fei and J. Tu, “Complete characterization of bicyclic graphs with the maximum and second-maximum degree Kirchhoff index,” Applied Mathematics and Computation, vol. 330, pp. 118–124, 2018. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. J. Liu, J. Zhao, S. Wang, M. Javaid, and J. Cao, “On the Topological Properties of the Certain Neural Networks,” Journal of Artificial Intelligence and Soft Computing Research, vol. 8, no. 4, pp. 257–268, 2018. View at Publisher · View at Google Scholar
  6. B. Ning and B. Li, “Spectral radius and traceability of connected claw-free graphs,” Filomat, vol. 30, pp. 2445–2452, 2016. View at Google Scholar
  7. Y. Wu and Y. Chen, “On the extremal eccentric connectivity index of graphs,” Applied Mathematics and Computation, vol. 331, pp. 61–68, 2018. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. C. Wang, G. Yu, W. Sun, and J. Cao, “The Least Eigenvalue of the Graphs Whose Complements Are Connected and Have Pendent Paths,” Journal of Artificial Intelligence and Soft Computing Research, vol. 8, no. 4, pp. 303–308, 2018. View at Publisher · View at Google Scholar
  9. D. Chen, The Harary index of a unicyclic graph [M.S. thesis], Hunan Normal University, Changsha, China, 2009.
  10. K. C. Das, B. Zhou, and N. Trinajstic, “Bounds on Harary index,” Journal of Mathematical Chemistry, vol. 46, no. 4, pp. 1377–1393, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  11. L. Feng and A. Ilić, “Zagreb, Harary and hyper-Wiener indices of graphs with a given matching number,” Applied Mathematics Letters, vol. 23, no. 8, pp. 943–948, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. S. He and S. Li, “On the signless Laplacian index of unicyclic graphs with fixed diameter,” Linear Algebra and its Applications, vol. 436, no. 1, pp. 252–261, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  13. A. Ilic, G. Yu, and L. Feng, “The Harary index of trees,” Utilitas Mathematica, vol. 87, pp. 21–31, 2012. View at Google Scholar · View at MathSciNet
  14. K. Xu and N. Trinajstić, “Hyper-Wiener and Harary indices of graphs with cut edges,” Utilitas Mathematica, vol. 84, pp. 153–163, 2011. View at Google Scholar · View at MathSciNet · View at Scopus
  15. K. Xu and K. C. Das, “Extremal unicyclic and bicyclic graphs with respect to Harary index,” Bulletin of the Malaysian Mathematical Sciences Society, vol. 36, no. 2, pp. 373–383, 2013. View at Google Scholar · View at MathSciNet
  16. Y. Nacaroglu and A. D. Maden, “On the eccentric connectivity index of unicyclic graphs,” Iranian Journal of Mathematical Chemistry, vol. 9, pp. 47–56, 2018. View at Google Scholar