  The eccentricity of a vertex is the maximum distance from it to another
vertex and the average eccentricity $ecc (G)$ of a graph $G$ is the mean value
of eccentricities of all vertices of $G$. The average eccentricity is deeply
connected with a topological descriptor called the eccentric connectivity
index, defined as a sum of products of vertex degrees and eccentricities. In
this paper we analyze extremal properties of the average eccentricity,
introducing two graph transformations that increase or decrease $ecc (G)$.
Furthermore, we resolve four conjectures, obtained by the system AutoGraphiX,
about the average eccentricity and other graph parameters (the clique number,
the Randi\' c index and the independence number), refute one AutoGraphiX
conjecture about the average eccentricity and the minimum vertex degree and
correct one AutoGraphiX conjecture about the domination number.
