On the Two Conjectures of the Wiener Index

The Wiener index of a graph, which is the sum of the distances between all pairs of vertices, has been well studied. Recently, Sills and Wang in 2012 proposed two conjectures on the maximal Wiener index of trees with a given degree sequence. This note proves one of the two conjectures and disproves the other.

[1]  Hua Wang,et al.  On the maximal Wiener index and related questions , 2012, Discret. Appl. Math..

[2]  Hua Wang,et al.  Corrigendum: The extremal values of the Wiener index of a tree with given degree sequence , 2007, Discret. Appl. Math..

[3]  Li-Qun Xu,et al.  The Wiener Index of Trees with Given Degree Sequences , 2008 .

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

[5]  Yong Liu,et al.  Maximum Wiener Index of Trees with Given Degree Sequence , 2010 .

[6]  Ronald L. Graham,et al.  On the addressing problem for loop switching , 1971 .

[7]  R. P. Kurshan,et al.  On the addressing problem of loop switching , 1972 .

[8]  Eberhard Triesch,et al.  Superdominance order and distance of trees with bounded maximum degree , 2003, Discret. Appl. Math..

[9]  Karen L. Collins.  On a conjecture of Graham and Lovász about distance matrices , 1989, Discret. Appl. Math..

[10]  David L. Powers,et al.  The distance spectrum of the path Pn and The First Distance Eigenvector of Connected Graphs , 1990 .

[11]  Dieter Rautenbach,et al.  Wiener index versus maximum degree in trees , 2002, Discret. Appl. Math..

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

[13]  Gerhard J. Woeginger,et al.  The Wiener maximum quadratic assignment problem , 2011, Discret. Optim..

[14]  Huiqing Liu,et al.  On the Wiener Index of Trees with Fixed Diameter , 2008 .

[15]  Elwood S. Buffa,et al.  Graph Theory with Applications , 1977 .

[16]  R. Graham,et al.  Distance Matrix Polynomials of Trees , 1978 .

[17]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[18]  Hua Wang,et al.  Corrigendum: The extremal values of the Wiener index of a tree with given degree sequence , 2009, Discret. Appl. Math..