The Wiener index of hypergraphs

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 $$1\le m\le n-1$$ 1 ≤ m ≤ 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.

[1]  Mikhail V. Goubko,et al.  Maximizing Wiener index for trees with given vertex weight and degree sequences , 2017, Appl. Math. Comput..

[2]  Kexiang Xu,et al.  A Survey on Graphs Extremal with Respect to Distance-Based Topological Indices , 2014 .

[3]  Elena V. Konstantinova,et al.  Application of hypergraph theory in chemistry , 2001, Discret. Math..

[4]  I. Gutman,et al.  Wiener Index of Hexagonal Systems , 2002 .

[5]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[6]  Philip B. Zhang,et al.  The Wiener index of Sierpiński-like graphs , 2018, J. Comb. Optim..

[7]  Shang-wang Tan.  The minimum Wiener index of unicyclic graphs with a fixed diameter , 2018 .

[8]  Elena V. Konstantinova,et al.  Molecular Hypergraphs: The New Representation of Nonclassical Molecular Structures with Polycentric Delocalized Bonds , 1995, J. Chem. Inf. Comput. Sci..

[9]  Hong Lin.  Extremal Wiener Index of Trees with Given Number of Vertices of Even Degree , 2014 .

[10]  Gholam Hossein Fath-Tabar.  Some New Upper Bounds on the Wiener and Edge Wiener Index of k-Connected Graphs , 2018, Ars Comb..

[11]  I. Gutman,et al.  Some recent results in the theory of the Wiener number , 1993 .

[12]  Claude Berge,et al.  Hypergraphs - combinatorics of finite sets , 1989, North-Holland mathematical library.

[13]  Shang-wang Tan,et al.  The largest Wiener index of unicyclic graphs given girth or maximum degree , 2017 .

[14]  Lin Sun,et al.  The Wiener Index of r-Uniform Hypergraphs , 2017 .

[15]  Kinkar Chandra Das,et al.  On maximum Wiener index of trees and graphs with given radius , 2017, J. Comb. Optim..

[16]  Shuchao Li,et al.  Cacti with n-vertices and t cycles having extremal Wiener index , 2017, Discret. Appl. Math..

[17]  Shang-wang Tan,et al.  The Wiener index of unicyclic graphs given number of pendant vertices or cut vertices , 2017 .

[18]  I. Gutman,et al.  Wiener Index of Trees: Theory and Applications , 2001 .