On Extremal Values Of Total Structure Connectivity and Narumi-Katayama Indices on the Class of all Unicyclic and Bicyclic Graphs

Document Type : Research Paper

Authors

Department of Mathematics, Shahrekord Branch, Islamic Azad University, Shahrekord, Iran

Abstract

‎The total structure connectivity and Narumi-Katayama indices of a simple graph $G$ are defined as $TS(G)={\prod_{{u}\in{V(G)}}}{\frac{1}{\sqrt {{d_{u}}}}}$ and $ NK(G)={\prod_{{u}\in{V(G)}}{{d_{u}}}}$ respectively‎, ‎where $d _{u} $ represents the degree of vertex $ u $ in $ G $‎. ‎In this paper‎, ‎we determine the extremal values of total structure connectivity index on the class of unicyclic and bicyclic graphs and characterize the corresponding extremal graphs‎. ‎In addition‎, ‎we determine the bicyclic graphs extremal with respect to the Narumi-Katayama index‎.

Keywords

Main Subjects

References

[1] G. H. Fath-Tabar, A. R. Ashrafi and D. Stevanović, Spectral properties of fullerenes, J. Comput. Theor. Nanosci. 9 (3) (2012) 327–329, https://doi.org/10.1166/jctn.2012.2027.
[2] O. C. Havare, QSPR analysis with curvilinear regression modeling and topological indices, Iranian J. Math. Chem. 10 (4) (2019) 331–341, https://doi.org/10.22052/IJMC.2019.191865.1448.
[3] H. Liu, L. You and Y. Huang, Ordering chemical graphs by Sombor indices and its applications, MATCH Commun. Math. Comput. Chem. 87 (1) (2022) 5–22, https://doi.org/10.46793/match.87-1.005L.
[4] F. Shafiei, Relationship between topological indices and thermodynamic properties and of the monocarboxylic acids applications in QSPR, Iranian J. Math. Chem. 6 (1) (2015) 15–28, https://doi.org/10.22052/IJMC.2015.8944.
[5] F. Taghvaee and G. H. Fath-Tabar, Relationship between coefficients of characteristic polynomial and matching polynomial of regular graphs and its applications, Iranian J. Math. Chem. 8 (1) (2017) 7–23, https://doi.org/10.22052/IJMC.2017.15093.
[6] Z. Yarahmadi and G. H. Fath-Tabar, The Wiener, Szeged, PI, vertex PI, the first and second Zagreb indices of N-branched phenylacetylenes dendrimers, MATCH Commun. Math. Comput. Chem. 65 (2011) 201–208.
[7] I. Gutman and N. Trinajstić, Graph theory and molecular orbital total ϕ-electron energy of alternant hydrocarbons, Chem. Phys. Lett. 17 (4) (1972) 535–538, https://doi.org/10.1016/0009-2614(72)85099-1.
[8] G. H. Fath-Tabar, Old and new Zagreb indices of graphs, MATCH Commun. Math. Comput. Chem. 65 (2011) 79–84.
[9] M. Črepnjak and P. Ž. Pleteršek, Correlation between heat of formation and fifth geometric–arithmetic index, Fuller. Nanotub. 27 (7) (2019) 559–565, https://doi.org/10.1080/1536383X.2019.1617278.
[10] H. Narumi and M. Katayama, Simple topological index: A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons, Mem. Fac. Eng. Hokkaido Univ. 16 (3) (1984) 209–214.
[11] M. Ghorbani, M. Songhori and I. Gutman, Modified Narumi-Katayama index, Kragujevac J. Sci. 34 (2012) 57–64.
[12] I. Gutman and M. Ghorbani, Some properties of the Narumi-Katayama index, Appl. Math. Lett. 25 (10) (2012) 1435–1438, https://doi.org/10.1016/j.aml.2011.12.018.
[13] V. Bozović, Z. K. Vukićević and G. Popivoda, Extremal values of total multiplicative sum Zagreb index and first multiplicative sum Zagreb coindex on unicyclic and bicyclic graphs, MATCH Commun. Math. Comput. Chem. 78 (2017) 417–430.
[14] M. A. Manian, S. Heidarian and F. khaksar Haghani, Maximum and minimum values of inverse degree and forgotten indices on the class of all unicyclic graphs, AKCE Int. J. graphs Comb. 20 (1) (2023) 57–60, https://doi.org/10.1080/09728600.2023.2170298.
[15] Z. You and B. Liu, On the extremal Narumi-Katayama index of graphs, Filomat 28 (3) (2014) 531–539.
[16] B. Wu and J. Meng, Basic properties of total transformation graphs, J. Math. Study 34 (2) (2001) 109–116.
[17] K. Xu and K. C. Das, Trees, unicyclic and bicyclic graphs extremal with respect to multiplicative sum Zagreb index, MATCH Commun. Math. Comput. Chem. 68 (2012) 257–272.
[18] K. Xu and I. Gutman, The greatest Hosoya index of bicyclic graphs with given maximum degree, MATCH Commun. Math. Comput. Chem. 66 (2011) 795–824
Iranian Journal of Mathematical Chemistry
Volume 14, Issue 3
September 2023
Pages 171-181

Files

Share

How to cite

Statistics