A family of graph theoretical descriptors for atoms and for molecules is introduced: the counts of all walks of increasing length (all walks as opposed to self-returning walks and to paths). These descriptors, while extremely easily calculated by a Morgan-type summation procedure, exhibit high discriminating power among isomers and among structural situations of atoms. A systematic search for degeneracies is performed within tree graphs. The walk-based topological index twe (total walk count) is less degenerate than Balaban's J, and a code for graphs based on all walks seems to be more discriminating than Trinajstić's OSC (ordered structural code). The new descriptors are successfully used as alternative variables in the description of ¹³C NMR data for usual and highly branched alkanes.