In this article, we develop a perturbative technique to construct families of non-isomorphic discrete graphs which are isospectral for the standard (also called normalised) Laplacian and its signless version. We use vertex contractions as a graph perturbation and spectral bracketing with auxiliary graphs which have certain eigenvalues with high multiplicity. There is no need to know explicitly the eigenfunctions of the corresponding graphs. In principle, one only needs to know the multiplicity of the eigenvalues of the auxiliary graphs, and that these eigenvalues are all different. We illustrate the method by presenting several families of examples of isospectral graphs including fuzzy balls, complete bipartite graphs and subdivision graphs obtained from the previous examples. All the examples constructed turn out to be also isospectral for the standard (Kirchhoff) Laplacian on the associated equilateral metric graph.