EXTREMAL HYPER ZAGREB INDEX
FOR TRICYCLIC GRAPHS
Zheng-Qing Chu1, Muhammad Kamran Jamil2
Aisha Javed3
1Department of General Education
Anhui Xinhua University, Hefei 230088, CHINA
2Department of Mathematics
Riphah Institute of Computing and Applied Sciences
Riphah International University, Lahore, PAKISTAN
3Abdus Salam School of Mathematical Sciences
Government College University, Lahore, PAKISTAN

Abstract

For a graph $G=(V(G),E(G))$, the first hyper Zagreb index is defined as $\sum_{uv\in E(G)}(d(u)+d(v))^2$, where $d(v)$ is the degree of the vertex $v$. The hyper Zagreb index is a kind of extensions of Zagreb index. In this paper, the monotonicity of the hyper Zagreb index under some graph transformations was studied. Using these mathematical properties, the extremal graph among tricyclic graphs are determined for hyper Zagreb index. Moreover, the sharp upper and lower bounds on the hyper Zagreb index of tricyclic graphs are provided.

Citation details of the article



Journal: International Journal of Applied Mathematics
Journal ISSN (Print): ISSN 1311-1728
Journal ISSN (Electronic): ISSN 1314-8060
Volume: 32
Issue: 6
Year: 2019

DOI: 10.12732/ijam.v32i6.5

Download Section



Download the full text of article from here.

You will need Adobe Acrobat reader. For more information and free download of the reader, please follow this link.

References

  1. [1] B. Bollobàs, P. Erdös, Graphs of extremal weights, Ars Combin., 50 (1998), 225-233.
  2. [2] J. Devillers, A.T. Balaban (Eds.), Topologicial Indices and Related De- scriptors in QSAR and QSPR, Gordon and Breach, Amsterdam (1999).
  3. [3] I. Gutman, K.C. Das, The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem., 50 (2004), 83-92.
  4. [4] K.C. Das, I. Gutman, Some properties of of the second Zagreb index, MATCH Commun. Math. Comput. Chem., 52 (2004), 103-112.
  5. [5] Y. Gao, M. Imran, M.R. Farahani, H.M.A. Siddiqui, Some connectivity indices and Zagreb index of honeycomb graphs, Int. J. Pharm. Sci. Res., 9, No 5 (2018), 2080-2087.
  6. [6] I. Gutman, M.K. Jamil, N. Akhter, Graphs with fixed number of pendent vertices and first Zagreb index, Trans. on Combinatorics, 4 (2015), 43-48.
  7. [7] W. Gao, M.K. Jamil, M.R. Farahani, The hyper Zagreb index and some graph operations, J. Appl. Math. Comput., 54, No 1-2 (2017), 263-275.
  8. [8] W. Gao, M.K. Jamil, A. Javed, M.R. Farahani, S. Wang, J.B. Liu, Sharp bounds of the hyper-Zagreb index on acylic, unicyclic and bicyclic graphs, Discrete Dynamics in Nature and Society, 2017 (2017), Article ID 6079450, 5 pp.; DOI: doi.org/10.1155/2017/6079450.
  9. [9] I. Gutman, N. Tirnajstic̀, Graph theory and molecular orbitals. III. Total pi-electron energy bydrocarbons, Chem. Phys. Lett., 17 (1972), 535-538.
  10. [10] G.H. Shirdel, H. Rezapor, A.M. Sayadi, The hyper-Zagreb index of graph operatiions, Iranian J. of Mathematical Chemistry, 4, No 2 (2013), 213- 220.
  11. [11] M.R. Farahani, The hyper-Zagreb index of Benzenoid series, Frontiers of Mathematics and Its Applications, 2, No 1 (2015), 1-5.
  12. [12] W. Gao, M.R. Farahani, The hyper-Zagreb index for an infinite family of nanostar dendrime, J. of Discrete Mathematical Sciences and Cryptogra- phy, 20), No 2 (2017), 515-523.
  13. [13] M.R. Farahani, W. Gao, M.R.R. Kanna, P.R. Kumar, J.B. Liu, General Randic, sum-connectivity, hyper-Zagreb and harmonic indices, and har- monic polynomial of molecular graphs, Advances in Physical Chemistry, 2016 (2016), Article ID 2315949, 6 pp., doi: 10.1155/2016/2315949.
  14. [14] M. Rezaei, W. Gao, M.K. Siddiqui, M.R. Farahani, Computing hyper Za- greb index and mpolynomials of titania nanotubes T io2 [m, n], Sigma J. of Engineering and Natural Sci., 35, No 4 (2017), 707-714.
  15. [15] S. Wang, W. Gao, M.K. Jamil, M.R. Farahani, J. Liu, Bounds of Zagreb indices and hyper Zagreb indices, Mathematical Reports, 21, No 71 (2019), 93-102.