Cycles, the Degree Distance, and the Wiener Index

Authors

Daniel Gray, University of FloridaFollow
Hua Wang, Georgia Southern UniversityFollow

Document Type

Article

Publication Date

10-2012

Publication Title

Open Journal of Discrete Mathematics

DOI

10.4236/ojdm.2012.24031

ISSN

2161-7643

Abstract

The degree distance of a graph G is D'(G)=(1/2)∑ⁿ_i=1∑ⁿ_j=1(d_i+d_j)L_{i ,j}, where d_i and d_j are the degrees of vertices v_i, v_j ∈ V (G), and L_i,j is the distance between them. The Wiener index is defined as W(G)=(1/2)∑ⁿi₌₁ ∑ⁿ_j-1L_{i, j}. An elegant result (Gutman; Klein, Mihalic, Plavsic and Trinajstic) is known regarding their correlation, that D'(T)=4W(T)-n(n-1)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.

Comments

Article under the Creative Commons license "Attribution" (CC BY). Article obtained from the Open Journal of Discrete Mathematics.

Recommended Citation

Gray, Daniel, Hua Wang. 2012. "Cycles, the Degree Distance, and the Wiener Index." Open Journal of Discrete Mathematics, 2 (4): 156-159. doi: 10.4236/ojdm.2012.24031
https://digitalcommons.georgiasouthern.edu/math-sci-facpubs/213