On the Difference Between the Eccentric Connectivity Index and Eccentric Distance Sum of Graphs

The eccentric connectivity index of a graph G is ξc(G)=∑v∈V(G)ε(v)deg(v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi ^c(G) = \sum _{v \in V(G)}\varepsilon (v)\deg (v)$$\end{document}, and the eccentric distance sum is ξd(G)=∑v∈V(G)ε(v)D(v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi ^d(G) = \sum _{v \in V(G)}\varepsilon (v)D(v)$$\end{document}, where ε(v)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon (v)$$\end{document} is the eccentricity of v, and D(v) the sum of distances between v and the other vertices. A lower and an upper bound on ξd(G)-ξc(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi ^d(G) - \xi ^c(G)$$\end{document} is given for an arbitrary graph G. Regular graphs with diameter at most 2 and joins of cocktail-party graphs with complete graphs form the graphs that attain the two equalities, respectively. Sharp lower and upper bounds on ξd(T)-ξc(T)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi ^d(T) - \xi ^c(T)$$\end{document} are given for arbitrary trees. Sharp lower and upper bounds on ξd(G)+ξc(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi ^d(G)+\xi ^c(G)$$\end{document} for arbitrary graphs G are also given, and a sharp lower bound on ξd(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi ^d(G)$$\end{document} for graphs G with a given radius is proved.