With the help of the AutoGraphiX system, we study relations of the form

where and are invariants of the graph , is one of the operations , and are best possible lower and upper bounding functions depending only one the size m of G. Specifically, we consider couples of indices where is a measure of distance, i.e., diameter, radius or average eccentricity, and is a measure of connectivity, i.e., minimum degree, edge connectivity and vertex connectivity. Conjectures are obtained and then proved in almost all cases.