Lower and Upper Bounds for Some Degree-Based Indices of Graphs

Year 2025, Volume: 8 Issue: 1, 12 - 18, 31.03.2025

Gül Özkan Kızılırmak , Emre Sevgi , Şerife Büyükköse , İsmail Naci Cangül

https://doi.org/10.33401/fujma.1366063

Abstract

Topological indices are mathematical measurements regarding the chemical structures of any simple finite graph. These are used for QSAR and QSPR studies. We get bounds for some degree based topological indices of a graph using solely the vertex degrees. We obtain upper and lower bounds for these indices and investigate for the complete graphs, path graphs and Fibonacci-sum graphs.

Keywords

Graph , Fibonacci-sum graph , Topological graph index

References

• [1] X. Li and I. Gutman, Mathematical Aspects of Randic-Type Molecular Structure Descriptors, Mathematical Chemistry Monographs, 1(1), Faculty of Science, University of Kragujevac, Kragujevac, (2006).
• [2] H. Narumi and M. Katayama, Simple topological index. A newly devised index characterizing the topological nature of structural isomers of saturated hydrocarbons, Mem. Fac. Engin. Hokkaido Univ., 16 (1984), 209-214. $\href{https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/38010/1/16(3)_209-214.pdf}{\mbox{[Web]}} $
• [3] D. Vukicevic and M. Gasperov, Bond additive modeling 1.Adriatic indices, Croat. Chem. Acta, 83(3) (2010), 243-260. $ \href{chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://citeseerx.ist.psu.edu/document?repid=rep1&type=pdf&doi=ee388e313c250ec065ed20ea9d45dbd58f9b8c65}{\mbox{[Web]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-78650489346&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Bond+Additive+Modeling+1.+Adriatic+Indices%22%29&sessionSearchId=2b3b881954449b5ab9e76e44896e2fdc}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000285799300001}{\mbox{[Web of Science]}} $
• [4] M. Bhanumathi and K.E.J. Rani, On multiplicative sum connectivity index, multiplicative Randic index and multiplicative harmonic index of some nanostar dendrimers, Int. J. Eng. Sci. Adv. Comput. Bio-Tech., 9(2) (2018), 52-67. $ \href{https://doi.org/10.26674/ijesacbt/2018/49410 }{\mbox{[CrossRef]}} $
• [5] I. Gutman and M. Ghorbani, Some properties of the Narumi–Katayama index, Appl. Math. Lett., 25(10) (2012), 1435–1438. $ \href{https://doi.org/10.1016/j.aml.2011.12.018}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84862998776&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Some+properties+of+the+Narumi%E2%80%93Katayama+index%22%29&sessionSearchId=2b3b881954449b5ab9e76e44896e2fdc}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000306872400036}{\mbox{[Web of Science]}} $
• [6] M. Ghorbani, M. Songhori and I. Gutman, Modified Narumi – Katayama index, Kragujevac J. Sci., 34 (2012), 57–64. $\href{https://www.pmf.kg.ac.rs/KJS/images/volumes/vol34/kjs34ghorbanigutman57.pdf}{\mbox{[Web]}} $
• [7] K. Ch. Das, M. Matejic, E. Milovanovic and I. Milovanovic, Bounds for symmetric division deg index of graphs, Filomat, 33(3) (2019), 683-698. $ \href{https://doi.org/10.2298/FIL1903683D}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85077850354&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Bounds+for+symmetric+division+deg+index+of+graphs%22%29&sessionSearchId=2b3b881954449b5ab9e76e44896e2fdc&relpos=1}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000496191500003}{\mbox{[Web of Science]}} $
• [8] K. Fox, W. B. Kinnersley, D. McDonald, N. Orlow and G. J. Puleo, Spanning paths in Fibonacci-Sum graphs, Fib. Quart.,52(1) (2014), 46-49. $\href{chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://www.fq.math.ca/Papers1/52-1/FoxKinnersleyMcDonaldOrlowPuleo.pdf}{\mbox{[Web]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-84896976822&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Spanning+paths+in+Fibonacci-Sum+graphs%22%29&sessionSearchId=2b3b881954449b5ab9e76e44896e2fdc}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000217593600008}{\mbox{[Web of Science]}} $
• [9] A. Arman, D.S. Gunderson and P.C. Li, Properties of the Fibonacci-sum graph, arXiv:1710.10303v1[math.CO] (2017). $ \href{http://dx.doi.org/10.48550/arXiv.1710.10303}{\mbox{[CrossRef]}} $
• [10] D. Tasçi, G. Özkan Kizilirmak, E. Sevgi and Ş Büyükköse, The bounds for the largest eigenvalues of Fibonacci-sum and Lucas-sum graphs, TWMS J. App. Eng. Math., 12(1) (2022), 367-371. $ \href{chrome-extension://efaidnbmnnnibpcajpcglclefindmkaj/https://belgelik.isikun.edu.tr/xmlui/bitstream/handle/iubelgelik/3414/vol.12.no.1-32.pdf?sequence=1&isAllowed=y}{\mbox{[Web]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85123522161&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22The+bounds+for+the+largest+eigenvalues+of+Fibonacci-sum+and+Lucas-sum+graphs%22%29&sessionSearchId=2b3b881954449b5ab9e76e44896e2fdc}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000739899800032}{\mbox{[Web of Science]}} $
• [11] A.Y. Güneş, S. Delen, M. Demirci, A.S. Çevik and İ.N. Cangül, Fibonacci Graphs, Symmetry, 12(9) (2020), 1383. $ \href{https://doi.org/10.3390/sym12091383}{\mbox{[CrossRef]}} \href{https://www.scopus.com/record/display.uri?eid=2-s2.0-85090400401&origin=resultslist&sort=plf-f&src=s&sot=b&sdt=b&s=TITLE-ABS-KEY%28%22Fibonacci+Graphs%22%29&sessionSearchId=2b3b881954449b5ab9e76e44896e2fdc&relpos=11}{\mbox{[Scopus]}} \href{https://www.webofscience.com/wos/woscc/full-record/WOS:000587623100001}{\mbox{[Web of Science]}} $