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 OJDM  Vol.2 No.4 , October 2012
Cycles, the Degree Distance, and the Wiener Index
Abstract: The degree distance of a graph G is , where and are the degrees of vertices , and is the distance between them. The Wiener index is defined as . An elegant result (Gutman; Klein, Mihali?,, Plav?i? and Trinajsti?) is known regarding their correlation, that for a tree T with n vertices. In this note, we extend this study for more general graphs that have frequent appearances in the study of these indices. In particular, we develop a formula regarding their correlation, with an error term that is presented with explicit formula as well as sharp bounds for unicyclic graphs and cacti with given parameters.
Cite this paper: D. Gray and H. Wang, "Cycles, the Degree Distance, and the Wiener Index," Open Journal of Discrete Mathematics, Vol. 2 No. 4, 2012, pp. 156-159. doi: 10.4236/ojdm.2012.24031.
References

[1]   H. Wiener, “Structural Determination of Paraffin Boiling Points,” Journal of the American Chemical Society, Vol. 69, No. 1, 1947, pp. 17-20. doi:10.1021/ja01193a005

[2]   A. A. Dobrynin and A. A. Kochetova, “Degree Distance of A Graph: A Degree Analogue of the Wiener Index,” Journal of Chemical Information and Computer Sciences, Vol. 34, No. 5, 1994, pp. 1082-1086. doi:10.1021/ci00021a008

[3]   P. Dankelmann, I. Gutman, S. Mukwembi and H. C. Swart, “On the Degree Distance of a Graph,” Discrete Applied Mathematics, Vol. 157, No. 13, 2009, pp. 2773-2777. doi:10.1016/j.dam.2009.04.006

[4]   A. A. Dobrynin, R. Entringer and I. Gutman, “Wiener Index of Trees: Theory and Applications,” Acta Applicandae Mathematicae, Vol. 66, No. 3, 2001, pp. 211-249. doi:10.1023/A:1010767517079

[5]   Z. Du and B. Zhou, “Minimum Wiener Indices of Trees and Unicyclic Graphs of Given Matching Number,” Match, Vol. 63, No. 1, 2010, pp. 101-112.

[6]   M. Fischermann, A. Hoffmann, D. Rautenbach, L. A. Székely and L. Volkmann, “Wiener Index versus Maximum Degree in Trees,” Discrete Applied Mathematics, Vol. 122, No. 1-3, 2002, pp. 127-137. doi:10.1016/S0166-218X(01)00357-2

[7]   H. Liu and M. Lu, “A Unified Approach to Cacti for Different Indices,” Match, Vol. 58, No. 1, 2007, pp. 183-194.

[8]   A. I. Tomescu, “Properties of Connected Graphs Having Minimum Degree Distance,” Discrete Applied Mathematics, Vol. 309, No. 9, 2009, pp. 2745-2748. doi:10.1016/j.disc.2008.06.031

[9]   A. I. Tomescu, “Unicyclic and Bicyclic Graphs Having Minimum Degree Distance,” Discrete Applied Mathematics, Vol. 156, No. 2, 2008, pp. 125-130. doi:10.1016/j.dam.2007.09.010

[10]   I. Gutman, “Selected Properties of the Schultz Molecular Topological Index,” Journal of Chemical Information and Computer Sciences, Vol. 34, No. 5, 1994, pp. 1087-1089. doi:10.1021/ci00021a009

[11]   D. J. Klein, Z. Mihalic′, D. Plav?ic′ and N. Trinajstic′, “Molecular Topological Index: A Relation with the Wiener Index,” Journal of Chemical Information and Computer Sciences, Vol. 32, No. 4, 1992, pp. 304-305.

 
 
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