We propose a universal approach to a range of enumeration problems in graphs by means of tensor networks. The key point is in contracting suitably chosen symmetric tensors placed at the vertices of a graph along the edges. This approach leads to simple formulas that count, in particular, the number of d-regular subgraphs of an arbitrary graph (including the number of d-factors) and of proper edge colorings. We briefly discuss the computational complexity of the algorithms based on these formulas.