On the difference between the eccentric connectivity index and eccentric distance sum of graphs

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.