Skip to main content
Springer Nature Link
Log in

On the Degree Distance of Unicyclic Graphs with Given Matching Number

  • Original Paper
  • Published:
Graphs and Combinatorics Aims and scope Submit manuscript

Abstract

Let \(G=(V_G, E_G)\) be a connected graph. The degree distance of \(G\) is defined as \(D'(G)= \sum _{\{u,v\} \subseteq V_G}(d_G(v)+d_G(v))d_G(u,v)\), where \(d_G(v)\) is the degree of vertex \(v, d_G(u, v)\) denotes the distance between \(u\) and \(v\) in \(G\). This parameter was introduced, independently, by Dobrynin and Kochetova and by Gutman as a weighted version of the Wiener index. Feng et al. (Graphs Comb 29(3):449–462, 2013) characterized \(n\)-vertex unicyclic graphs with given matching number having the minimal degree distance. As a continuance of it, in this paper the \(n\)-vertex unicyclic graphs of given matching number with the second and third minimal degree distance are identified respectively.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
€32.70 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (France)

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. Macmillan Press, New York (1976)

    Book  MATH  Google Scholar 

  2. Bucicovschi, O., Cioabǎ, S.M.: The minimum degree distance of graphs of given order and size. Discrete Appl. Math. 156, 3518–3521 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Chang, A., Tian, F.: On the spectral radius of unicyclic graphs with perfect matching. Linear Algebra Appl. 370, 237–250 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Dobrynin, A.A., Kochetova, A.A.: Degree distance of a graph: a degree analogue of the Wiener index. J. Chem. Inf. Comput. Sci. 34, 1082–1086 (1994)

    Article  Google Scholar 

  5. Dankelmann, P., Gutman, I., Mukwembi, S., Swart, H.C.: On the degree distance of a graph. Discrete Appl. Math. 157, 2773–2777 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. Diudea, M.V., Gutman, I., Jäntschi, L.: Molecular Topology. Nova, Huntington (2001)

    Google Scholar 

  7. Dobrynin, A., Entringer, R., Gutman, I.: Wiener index of trees: theory and applications. Acta Appl. Math. 66, 211–249 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  8. Feng, L.H., Liu, W.J., Ilić, A., Yu, G.H.: Degree distance of unicyclic graphs with given matching number. Graphs Comb. 29(3), 449–462 (2013)

    Article  MATH  Google Scholar 

  9. Gutman, I., Polansky, O.E.: Mathematical Concepts in Organic Chemistry. Springer, Berlin (1986)

    Book  MATH  Google Scholar 

  10. Gutman, I.: Selected properties of the Schultz molecular topological index. J. Chem. Inf. Comput. Sci. 34, 1087–1089 (1994)

    Article  Google Scholar 

  11. Hou, Y.P., Li, J.S.: Bounds on the largest eigenvalues of trees with a given size of matching. Linear Algebra Appl. 342, 203–217 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ilić, A., Stevanović, D., Feng, L.H., Yu, G.H., Dankelmann, P.: Degree distance of unicyclic and bicyclic graphs. Discrete Appl. Math. 159, 779–788 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Ilić, A., Klavžar, S., Stevanović, D.: Calculating the degree distance of partial Hamming graphs. MATCH Commun. Math. Comput. Chem. 63, 411–424 (2010)

    MathSciNet  MATH  Google Scholar 

  14. B. McKay, Nauty. http://cs.anu.edu.au/bdm/nauty/

  15. Todeschini, R., Consonni, V.: Handbook of Molecular Descriptors. Wiley-VCH, Weinheim (2000)

    Book  Google Scholar 

  16. Tomescu, I.: Some extremal properties of the degree distance of a graph. Discrete Appl. Math. 98, 159–163 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Tomescu, I.: Properties of connected graphs having minimum degree distance. Discrete Math. 309, 2745–2748 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  18. Tomescu, I.: Unicyclic and bicyclic graphs having minimum degree distance. Discrete Appl. Math. 156, 125–130 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  19. Tomescu, I.: Ordering connected graphs having small degree distances. Discrete Appl. Math. 158, 1714–1717 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Tomescu, I., Kanwal, S.: Ordering connected graphs having small degree distances. II. MATCH Commun. Math. Comput. Chem. 67, 425–437 (2012)

    MathSciNet  MATH  Google Scholar 

  21. Wiener, H.: Structural determination of paraffin boiling point. J. Amer. Chem. Soc. 69, 17–20 (1947)

    Article  Google Scholar 

  22. Yu, A., Tian, F.: On the spectral radius of unicyclic graphs. MATCH Commun. Math. Comput. Chem. 51, 97–109 (2004)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to express their sincere gratitude to both of the referees for a very careful reading of this paper and for all their insightful comments, which led to number of improvements to this paper. This work is financially supported by the National Natural Science Foundation of China (Grant Nos. 11271149,11371162), the Program for New Century Excellent Talents in University (Grant No. NCET-13-0817) and the Special Fund for Basic Scientific Research of Central Colleges (Grant No. CCNU13F020).

Author information

Authors and Affiliations

  1. Faculty of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, People’s Republic of China

    Shuchao Li, Yibing Song & Huihui Zhang

Authors
  1. Shuchao Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yibing Song
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Huihui Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shuchao Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, S., Song, Y. & Zhang, H. On the Degree Distance of Unicyclic Graphs with Given Matching Number. Graphs and Combinatorics 31, 2261–2274 (2015). https://doi.org/10.1007/s00373-015-1527-4

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00373-015-1527-4

Keywords

Mathematics Subject Classification

Search

Navigation