  The eccentric connectivity index, proposed by Sharma, Goswami and Madan, has
been employed successfully for the development of numerous mathematical models
for the prediction of biological activities of diverse nature. We now report
mathematical properties of the eccentric connectivity index. We establish
various lower and upper bounds for the eccentric connectivity index in terms of
other graph invariants including the number of vertices, the number of edges,
the degree distance and the first Zagreb index. We determine the n-vertex trees
of diameter with the minimum eccentric connectivity index, and the n-vertex
trees of pendent vertices, with the maximum eccentric connectivity index. We
also determine the n-vertex trees with respectively the minimum, second-minimum
and third-minimum, and the maximum, second-maximum and third-maximum eccentric
connectivity indices for
