  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.
