We present an algorithm for constructing efficient topological shape descriptors of three dimensional objects. Given a smooth surface S and a Morse function f defined on S, our algorithm encodes the relationship among the critical points of the function f by means of a connection graph, called the Morse Connections Graph, whose nodes represent the critical points of f. Two nodes are related by an edge if a connection is established between them. This graph structure is extremely suitable for shape comparison and shape matching and inherits the invariant properties of the given Morse function f.