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On the extremal graphs with respect to the total reciprocal edge-eccentricity

  • Lifang Zhao
  • Hongshuai Li
  • Yuping GaoEmail author
Article
  • 37 Downloads

Abstract

The total reciprocal edge-eccentricity of a graph G is defined as \(\xi ^{ee}(G)=\sum _{u\in V_G}\frac{d_G(u)}{\varepsilon _G(u)}\), where \(d_G(u)\) is the degree of u and \(\varepsilon _G(u)\) is the eccentricity of u. In this paper, we first characterize the unique graph with the maximum total reciprocal edge-eccentricity among all graphs with a given number of cut vertices. Then we determine the k-connected bipartite graphs of order n with diameter d having the maximum total reciprocal edge-eccentricity. Finally, we identify the unique tree with the minimum total reciprocal edge-eccentricity among the n-vertex trees with given degree sequence.

Keywords

Total reciprocal edge-eccentricity Cut vertex Eccentricity Degree sequence 

Mathematics Subject Classification

05C69 05C05 

Notes

Acknowledgements

The authors wish to thank the anonymous referees for their careful reading and valuable comments on how to improve this paper.

References

  1. Behmaram A, Yousefi-Azari H, Ashrafi AR (2012) Wiener polarity index of fullerenes and hexagonal systems. Appl Math Lett 25(10):1510–1513MathSciNetzbMATHCrossRefGoogle Scholar
  2. Bondy JA, Murty USR (2008) Graph theory. In: Axler S, Ribet KA (eds) Graduate texts in mathematics, vol 244. Springer, New YorkGoogle Scholar
  3. Dankelmann P, Goddard W, Swart CS (2004) The average eccentricity of a graph and its subgraphs. Util Math 65:41–51MathSciNetzbMATHGoogle Scholar
  4. Dobrynin A, Kochetova AA (1994) Degree distance of a graph: a degree analogue of the Wiener index. J Chem Inf Comput Sci 34:1082–1086CrossRefGoogle Scholar
  5. Dobrynin AA, Entriger R, Gutman I (2001) Wiener index of trees: theory and applications. Acta Appl Math 66:211–249MathSciNetzbMATHCrossRefGoogle Scholar
  6. Dobrynin AA, Gutman I, Klavžar S, Žigert P (2002) Wiener index of hexagonal systems. Acta Appl Math 72:247–294MathSciNetzbMATHCrossRefGoogle Scholar
  7. Entringer RC, Jackson DE, Snyder DA (1976) Distance in graphs. Czech Math J 26:283–296MathSciNetzbMATHGoogle Scholar
  8. Geng XY, Li SC, Zhang M (2013) Extremal values on the eccentric distance sum of trees. Discrete Appl Math 161:2427–2439MathSciNetzbMATHCrossRefGoogle Scholar
  9. Georgakopoulos A, Wagner S (2017) Hitting times, cover cost, and the Wiener index of a tree. J Graph Theory 84(3):311–326MathSciNetzbMATHCrossRefGoogle Scholar
  10. Gupta S, Singh M, Madan AK (2000) Connective eccentricity index: a novel topological descriptor for predicting biological activity. J Mol Graph Model 18:18–25CrossRefGoogle Scholar
  11. Gupta S, Singh M, Madan AK (2002a) Application of graph theory: relationship of eccentric connectivity index and Wiener’s index with anti-inflammatory activity. J Math Anal Appl 266:259–268MathSciNetzbMATHCrossRefGoogle Scholar
  12. Gupta S, Singh M, Madan AK (2002b) Eccentric distance sum: a novel graph invariant for predicting biological and physical properties. J Math Anal Appl 275(1):386–401MathSciNetzbMATHCrossRefGoogle Scholar
  13. Gutman I (1994) Selected properties of the Schultz molecular topological index. J Chem Inf Comput Sci 34:1087–1089CrossRefGoogle Scholar
  14. Gutman I, Wagner S (2012) The matching energy of a graph. Discrete Appl Math 160(15):2177–2187MathSciNetzbMATHCrossRefGoogle Scholar
  15. Hosoya H (1971) Topological index. Newly proposed quautity characterizing the topological nature of structure of isomers of saturated hydrocarbons. Bull Chem Soc Jpn 44:2332–2337CrossRefGoogle Scholar
  16. Hosoya H (1972) Topological index as strong sorting device for coding chemical structure. J Chem Doc 12:181–183CrossRefGoogle Scholar
  17. Hou AL, Li SC, Song LZ, Wei B (2011) Sharp bounds for Zagreb indices of maximal outerplanar graphs. J Comb Optim 22(2):252–269MathSciNetzbMATHCrossRefGoogle Scholar
  18. Huang J, Li SC, Li XC (2016a) The normalized Laplacian, degree-Kirchhoff index and spanning trees of the linear polyomino chains. Appl Math Comput 289:324–334MathSciNetzbMATHGoogle Scholar
  19. Huang J, Li SC, Sun LQ (2016b) The normalized Laplacians, degree-Kirchhoff index and the spanning trees of linear hexagonal chains. Discrete Appl Math 207:67–79MathSciNetzbMATHCrossRefGoogle Scholar
  20. Ilić A (2012) On the extremal properties of the average eccentricity. Comput Math Appl 64:2877–2885MathSciNetzbMATHCrossRefGoogle Scholar
  21. Knor M, Škrekovski R, Tepeh A (2016) Mathematical aspects of Wiener index. Ars Math Contemp 11:327–352MathSciNetzbMATHCrossRefGoogle Scholar
  22. Kumar V, Sardana S, Madan AK (2004) Predicting anti-HIV activity of 2,3-diaryl-1,3 thiazolidin-4-ones: computational approach using reformed eccentric connectivity index. J Mol Model 10:399–407CrossRefGoogle Scholar
  23. Li SC (2017) Sharp bounds on the eccentric distance sum of graphs. In: Gutman I, Furtula B, Das KC, Milovanović E, Milovanović I (eds) Bounds in chemical graph theory-mainstreams. University Kragujevac, Kragujevac, pp 207–237Google Scholar
  24. Li XL, Shi YT (2008) A survey on the Randić index. MATCH Commun Math Comput Chem 59:127–156MathSciNetzbMATHGoogle Scholar
  25. Li SC, Song YB (2014) On the sum of all distances in bipartite graphs. Discrete Appl Math 169:176–185MathSciNetzbMATHCrossRefGoogle Scholar
  26. Li SC, Wu YY (2016a) On the extreme eccentric distance sum of graphs with some given parameters. Discrete Appl Math 206:90–99MathSciNetzbMATHCrossRefGoogle Scholar
  27. Li SC, Zhao LF (2016b) On the extremal total reciprocal edge-eccentricity of trees. J Math Anal Appl 433:587–602MathSciNetzbMATHCrossRefGoogle Scholar
  28. Li SC, Zhang HH (2017) Proofs of three conjectures on the quotients of the (revised) Szeged index and the Wiener index and beyond. Discrete Math 340(3):311–324MathSciNetzbMATHCrossRefGoogle Scholar
  29. Li SC, Zhang M, Yu GH, Feng LH (2012) On the extremal values of the eccentric distance sum of trees. J Math Anal Appl 390:99–112MathSciNetzbMATHCrossRefGoogle Scholar
  30. Li SC, Wu YY, Sun LL (2015) On the minimum eccentric distance sum of bipartite graphs with some given parameters. J Math Anal Appl 430:1149–1162MathSciNetzbMATHCrossRefGoogle Scholar
  31. Li HS, Li SC, Zhang HH (2017) On the maximal connective eccentricity index of bipartite graphs with some given parameters. J Math Anal Appl 454:453–467MathSciNetzbMATHCrossRefGoogle Scholar
  32. Liu RF, Du X, Jia HC (2016) Wiener index on traceable and Hamiltonian graphs. Bull Aust Math Soc 94:362–372MathSciNetzbMATHCrossRefGoogle Scholar
  33. Mohar B (2015) Median eigenvalues and the HOMO-LUMO index of graphs. J Combin Theory Ser B 112:78–92MathSciNetzbMATHCrossRefGoogle Scholar
  34. Nikiforov V, Agudelo N (2017) On the minimum trace norm/energy of \((0,1)\)-matrices. Linear Algebra Appl 526:42–59MathSciNetzbMATHCrossRefGoogle Scholar
  35. Sedlar J (2012) On augmented eccentric connectivity index of graphs and trees. MATCH Commun Math Comput Chem 68(1):325–342MathSciNetzbMATHGoogle Scholar
  36. Sharma V, Goswami R, Madan AK (1997) Eccentric connectivity index: a novel highly discriminating topological descriptor for structure property and structure activity studies. J Chem Inf Comput Sci 37:273–282CrossRefGoogle Scholar
  37. Tang L, Wang X, Liu WJ, Feng LH (2017) The extremal values of connective eccentricity index for trees and unicyclic graphs. Int J Comput Math 94(3):437–453MathSciNetzbMATHCrossRefGoogle Scholar
  38. Wang X, Tang L, Chen XS, Li MS, Li Y (2018) On the connective eccentricity index of graphs with fixed clique number. Ars Combin 138:105–117MathSciNetzbMATHGoogle Scholar
  39. Wiener H (1947a) Correlation of heat of isomerization and difference in heat of vaporization of isomers among paraffin hydrocaibons. J Am Chem Soc 69:2636–2638CrossRefGoogle Scholar
  40. Wiener H (1947b) Influence of interatomic forces on paraffin properties. J Chem Phys 15:766–767CrossRefGoogle Scholar
  41. Wiener H (1947c) Structural determination of paraffin boiling points. J Am Chem Soc 69:17–20CrossRefGoogle Scholar
  42. Wiener H (1948) Vapour–pressure–temperature relations among the branched paraffin hydrocarbons. J Chem Phys 15:425–430CrossRefGoogle Scholar
  43. Xu KX (2017) Some bounds on the eccentricity-based topological indices of graphs. In: Gutman I, Furtula B, Das KC, Milovanović E, Milovanović I (eds) Bounds in chemical graph theory-mainstreams. University Kragujevac, Kragujevac, pp 189–205Google Scholar
  44. Xu KX, Das KCh, Liu HQ (2016) Some extremal results on the connective eccentricity index of graphs. J Math Anal Appl 433(2):803–817MathSciNetzbMATHCrossRefGoogle Scholar
  45. Yu GH, Feng LH (2013) On connective eccentricity index of graphs. MATCH Commun Math Comput Chem 69:611–628MathSciNetzbMATHGoogle Scholar
  46. Yu GH, Qu H, Tang L, Feng LH (2014) On connective eccentricity index of trees and unicyclic graphs with given diameter. J Math Anal Appl 420(2):1776–1786MathSciNetzbMATHCrossRefGoogle Scholar
  47. Zhang HH, Chen J, Li SC (2017) On the quotients between the (revised) Szeged index and Wiener index of graphs. Discrete Math Theor Comput Sci 19, no. 1, Paper No. 12Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsLanzhou UniversityLanzhouPeople’s Republic of China
  2. 2.Zhongshan Overseas Chinese Secondary SchoolZhongshanPeople’s Republic of China

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