Skip to main content

On the number of connected subgraphs of graphs

Indian Journal of Pure and Applied Mathematics volume 52, pages 571–583 (2021)Cite this article

Abstract

For a connected graph G,  we denote the number of connected subgraphs of G by F(G). For a tree T, F(T) has been studied extensively and it has been observed that F(T) has a reverse correlation with Wiener index of T. We call F(G),  the subgraph index of G. In this paper, we study the subgraph index of unicyclic graphs and graphs with fixed number of pendant vertices. We obtain the unicyclic graphs which extremize the subgraph index over all unicyclic graphs on n vertices. The graphs which extremize the subgraph index among all unicyclic graphs with fixed girth are also obtained. Among all connected graphs on n vertices with fixed number of pendant vertices, the graph which minimizes and the graph which maximizes the subgraph index are characterized.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1

References

  1. Z. Chen, The number of subtrees of trees with given diameter, Discrete Math. Algorithms Appl., 10 (2018), no. 1, 1850001, 16 pp.

  2. R. Kirk and H. Wang, Largest number of subtrees of trees with a given maximum degree, SIAM J. Discrete Math., 22 (2008), no. 3, 985–995.

    MathSciNet  Article  Google Scholar 

  3. S. Li and S. Wang, Further analysis on the total number of subtrees of trees, Electron. J. Combin., 19 (2012), no. 4, P48, 14 pp.

  4. A. V. Sills and H. Wang, The minimal number of subtrees of a tree, Graphs Combin., 31 (2015), no. 1, 255–264.

    MathSciNet  Article  Google Scholar 

  5. L. A. Székely and H. Wang, On subtrees of trees, Adv. in Appl. Math., 34 (2005), no. 1, 138–155.

    MathSciNet  Article  Google Scholar 

  6. L. A. Székely and H. Wang, Binary trees with the largest number of subtrees, Discrete Appl. Math., 155 (2007), no. 3, 374–385.

    MathSciNet  Article  Google Scholar 

  7. H. S. Wilf, Generatingfunctionology, Second edition, Academic Press (1994).

    MATH  Google Scholar 

  8. Y. Xiao, H. Zhao, Z. Liu and Y. Mao, Trees with large numbers of subtrees, Int. J. Comput. Math., 94 (2017), no. 2, 372–385.

    MathSciNet  Article  Google Scholar 

  9. W. Yan and Y. N. Yeh, Enumeration of subtrees of trees, Theoret. Comput. Sci., 369 (2006), no. 1-3, 256–268.

    MathSciNet  Article  Google Scholar 

  10. G. Yu and L. Feng, On the Wiener index of unicyclic graphs with given girth, Ars Combin., 94 (2010), 361–369.

    MathSciNet  MATH  Google Scholar 

  11. X. M. Zhang, X. D. Zhang, D. Gray and H. Wang, Trees with the most subtrees - an algorithmic approach, J. Comb., 3 (2012), no. 2, 207–223.

    MathSciNet  MATH  Google Scholar 

  12. X. M. Zhang, X. D. Zhang, D. Gray and H. Wang, The number of subtrees of trees with given degree sequence, J. Graph Theory, 73 (2013), no. 3, 280–295.

    MathSciNet  Article  Google Scholar 

  13. X. M. Zhang and X. D. Zhang, The minimal number of subtrees with a given degree sequences, Graphs Combin., 31 (2015), no. 1, 309–318.

    MathSciNet  Article  Google Scholar 

  14. H. X. Zhao, Research on subgraph polynomials and reliability of networks, Ph.D. thesis, School of Computer, Northwestern Polytechnical University, Xi’an, China, 2004.

Download references

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, National Institute of Science Education and Research (NISER), P.O.- Jatni, District- Khurda, Bhubaneswar, Odisha, 752050, India

    Dinesh Pandey & Kamal Lochan Patra

  2. Homi Bhabha National Institute (HBNI), Training School Complex, Anushakti Nagar, Mumbai, 400094, India

    Dinesh Pandey & Kamal Lochan Patra

Authors
  1. Dinesh Pandey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kamal Lochan Patra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Kamal Lochan Patra.

Additional information

Communicated by Sharad S Sane, PhD.

D. Pandey:Supported by UGC Fellowship scheme (Sr. No. 2061641145), Government of India.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Pandey, D., Patra, K.L. On the number of connected subgraphs of graphs. Indian J Pure Appl Math 52, 571–583 (2021). https://doi.org/10.1007/s13226-021-00061-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13226-021-00061-4

Keywords

  • Tree
  • Unicyclic graph
  • Girth
  • Subtree core
  • Wiener index

Mathematics Subject Classification

  • 05C05
  • 05C07
  • 05C30
  • 05C35