Abstract
The eccentric connectivity index of a graph G is \(\xi ^c(G) = \sum _{v \in V(G)}\varepsilon (v)\deg (v)\), and the eccentric distance sum is \(\xi ^d(G) = \sum _{v \in V(G)}\varepsilon (v)D(v)\), where \(\varepsilon (v)\) 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 \(\xi ^d(G) - \xi ^c(G)\) 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 \(\xi ^d(T) - \xi ^c(T)\) are given for arbitrary trees. Sharp lower and upper bounds on \(\xi ^d(G)+\xi ^c(G)\) for arbitrary graphs G are also given, and a sharp lower bound on \(\xi ^d(G)\) for graphs G with a given radius is proved.
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Sandi Klavžar acknowledges the financial support from the Slovenian Research Agency (research core Funding P1-0297 and Projects J1-9109, J1-1693, N1-0095).
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Communicated by Rosihan M. Ali.
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Alizadeh, Y., Klavžar, S. On the Difference Between the Eccentric Connectivity Index and Eccentric Distance Sum of Graphs. Bull. Malays. Math. Sci. Soc. 44, 1123–1134 (2021). https://doi.org/10.1007/s40840-020-01015-5
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DOI: https://doi.org/10.1007/s40840-020-01015-5