Journal of Combinatorial Optimization

, Volume 39, Issue 2, pp 351–364 | Cite as

The Wiener index of hypergraphs

  • Xiangxiang Liu
  • Ligong WangEmail author
  • Xihe Li


The Wiener index is defined to be the sum of distances between every unordered pair of vertices in a connected hypergraph. In this paper, we first study how the Wiener index of a hypergraph changes under some graft transformations. For \(1\le m\le n-1\), we obtain the unique hypertree that achieves the minimum (or maximum) Wiener index in the class of hypertrees on n vertices and m edges. Then we characterize the unique hypertrees on n vertices with first three smallest Wiener indices, and the unique hypertree (not 2-uniform) with maximum Wiener index, respectively. In addition, we determine the unique hypergraph that achieves the minimum Wiener index in the class of hypergraphs on n vertices and p pendant edges.


Hypergraph Hypertree Wiener index 

Mathematics Subject Classification

05C50 05C65 



  1. Berge C (1989) Hypergraphs: combinatorics of finite sets. North-Holland, AmsterdamzbMATHGoogle Scholar
  2. Das KC, Nadjafi-Arani MJ (2017) On maximum Wiener index of trees and graphs with given radius. J Comb Optim 34:574–587MathSciNetCrossRefGoogle Scholar
  3. Dobrynin AA, Entringer R, Gutman I (2001) Wiener index of trees: theory and applications. Acta Appl Math 66:211–249MathSciNetCrossRefGoogle Scholar
  4. Dobrynin AA, Gutman I, Klavžar S, Žigert P (2002) Wiener index of hexagonal systems. Acta Appl Math 72:247–294MathSciNetCrossRefGoogle Scholar
  5. Fath-Tabar GH (2018) Some new upper bounds on the Wiener and edge Wiener index of \(k\)-connected graphs. ARS Comb 136:335–339MathSciNetzbMATHGoogle Scholar
  6. Guo HY, Zhou B, Lin HY (2017) The Wiener index of uniform hypergraphs. MATCH Commun Math Comput Chem 78:133–152MathSciNetGoogle Scholar
  7. Goubko M (2018) Maximizing Wiener index for trees with given vertex weight and degree sequences. Appl Math Comput 316:102–114MathSciNetzbMATHGoogle Scholar
  8. Gutman I, Li SC, Wei W (2017) Cacti with \(n\)-vertices and \(t\) cycles having extremal Wiener index. Discret Appl Math 232:189–200MathSciNetCrossRefGoogle Scholar
  9. Gutman I, Yeh YN, Lee SL, Luo YL (1993) Some recent results in the theory of the Wiener number. Indian J Chem 32A:651–661Google Scholar
  10. Konstantinova EV, Skorobogatov VA (1995) Molecular hypergraphs: the new representation of nonclassical molecular structures with polycentric delocalized bonds. J Chem Int Comput Sci 35:472–478CrossRefGoogle Scholar
  11. Konstantinova EV, Skorobogatov VA (2001) Application of hypergraph theory in chemistry. Discret Math 235:365–383MathSciNetCrossRefGoogle Scholar
  12. Knor M, Lužar B, Škrekovski R, Gutman I (2014) On Wiener index of common neighborhood graphs. MATCH Commun Math Comput Chem 72:321–332MathSciNetzbMATHGoogle Scholar
  13. Lin H (2014) Extremal Wiener index of trees with given number of vertices of even degree. MATCH Commun Math Comput Chem 72:311–320MathSciNetzbMATHGoogle Scholar
  14. Luo CM, Zuo LC, Zhang PB (2018) The Wiener index of Sierpiński-like graphs. J Comb Optim 35:814–841MathSciNetCrossRefGoogle Scholar
  15. Rodríguez JA (2005) On the Wiener index and the eccentric distance sum of hypergraphs. MATCH Commun Math Comput Chem 54:209–220MathSciNetzbMATHGoogle Scholar
  16. Sun L, Wu JL, Cai H, Luo ZY (2017) The Wiener index of \(r\)-uniform hypergraph. Bull Malays Math Sci Soc 40:1093–1113MathSciNetCrossRefGoogle Scholar
  17. Tan SW (2018) The minimum Wiener index of unicyclic graphs with a fixed diameter. J Appl Math Comput 56:93–114MathSciNetCrossRefGoogle Scholar
  18. Tan SW, Lin Y (2017) The largest Wiener index of unicyclic graphs given girth or maximum degree. J Appl Math Comput 53:343–363MathSciNetCrossRefGoogle Scholar
  19. Tan SW, Wang QL, Lin Y (2017) The Wiener index of unicyclic graphs given number of pendant vertices or cut vertices. J Appl Math Comput 55:1–24MathSciNetCrossRefGoogle Scholar
  20. Wiener H (1947) Structural determination of paraffin boiling points. J Am Chem Soc 69:17–20CrossRefGoogle Scholar
  21. Xu KX, Liu MH, Das KC, Gutman I, Furtula B (2014) A survey on graphs extremal with respect to distance-based topological indices. MATCH Commun Math Comput Chem 71:461–508MathSciNetzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of Applied Mathematics, School of ScienceNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China
  2. 2.Xi’an-Budapest Joint Research Center for CombinatoricsNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China

Personalised recommendations