On the difference between the Szeged and the Wiener index

We prove a conjecture of Nadjafi-Arani et al. on the difference between the Szeged and the Wiener index of a graph (Nadjafi-Aranifi et al., 2012). Namely, if G is a 2-connected non-complete graph on n vertices, then Sz(G)−W(G)≥2n−6. Furthermore, the equality is obtained if and only if G is the complete graph Kn−1 with an extra vertex attached to either 2 or n−2 vertices of Kn−1. We apply our method to strengthen some known results on the difference between the Szeged and the Wiener index of bipartite graphs, graphs of girth at least five, and the difference between the revised Szeged and the Wiener index. We also propose a stronger version of the aforementioned conjecture.

[1]  Ivan Gutman,et al.  Vertex Version of the Wiener Theorem , 2014 .

[2]  Xueliang Li,et al.  On a relation between Szeged and Wiener indices of bipartite graphs , 2012 .

[3]  Andrey A. Dobrynin,et al.  The Szeged Index and an Analogy with the Wiener Index , 1995, J. Chem. Inf. Comput. Sci..

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

[5]  Martin Knor,et al.  Wiener Index of Line Graphs , 2014 .

[6]  Xueliang Li,et al.  The (revised) Szeged index and the Wiener index of a nonbipartite graph , 2014, Eur. J. Comb..

[7]  Ali Reza Ashrafi,et al.  Graphs whose Szeged and Wiener numbers differ by 4 and 5 , 2012, Math. Comput. Model..

[8]  Yongtang Shi,et al.  On Wiener polarity index of bicyclic networks , 2016, Scientific Reports.

[9]  Roger C. Entringer,et al.  Distance in graphs , 1976 .

[10]  Riste Škrekovski,et al.  On Wiener Index of Common Neighborhood Graphs , 2014 .

[11]  Sandi Klavzar,et al.  Improved bounds on the difference between the Szeged index and the Wiener index of graphs , 2014, Eur. J. Comb..

[12]  Frank Harary,et al.  Status and Contrastatus , 1959 .

[13]  Ľubomír Šoltés,et al.  Transmission in graphs: A bound and vertex removing , 1991 .

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

[15]  Ali Reza Ashrafi,et al.  On the differences between Szeged and Wiener indices of graphs , 2011, Discret. Math..

[16]  Sandi Klavžar,et al.  The Szeged and the Wiener Index of Graphs , 1996 .

[17]  Hong Lin.  On the Wiener Index of Trees with Given Number of Branching Vertices , 2014 .

[18]  Sandi Klavzar,et al.  On the difference between the revised Szeged index and the Wiener index , 2014, Discret. Math..