For a connected locally path-connected topological space X and a continuous
function f on it such that its Reeb graph Rf is a finite topological graph,
we show that the cycle rank of Rf, i.e., the first Betti number b1(Rf), in
computational geometry called number of loops, is bounded from above by the
co-rank of the fundamental group ?1(X), the condition of local
path-connectedness being important since generally b1(Rf) can even exceed
b1(X). We give some practical methods for calculating the co-rank of ?1(X)
and a closely related value, the isotropy index. We apply our bound to
improve upper bounds on the distortion of the Reeb quotient map, and thus on
the Gromov-Hausdorff approximation of the space by Reeb graphs, for the
distance function on a compact geodesic space and for a simple Morse
function on a closed Riemannian manifold. This distortion is bounded from
below by what we call the Reeb width b(M) of a metric space M, which
guarantees that any real-valued continuous function on M has large enough
contour (connected component of a level set). We show that for a Riemannian
manifold, b(M) is non-zero and give a lower bound on it in terms of
characteristics of the manifold. In particular, we show that any real-valued
continuous function on a closed Euclidean unit ball E of dimension at least
two has a contour C with diam(C??E)??3.