For a connected locally path-connected topological space X and a continuous function f on it such that its Reeb graph R f is a finite topological graph, we show that the cycle rank of R f , i.e., the first Betti number b 1 (R f), in computational geometry called number of loops, is bounded from above by the co-rank of the fundamental group π 1 (X), the condition of local path-connectedness being important since generally b 1 (R f) can even exceed b 1 (X). We give some practical methods for calculating the co-rank of π 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.