This paper is devoted to the construction of Algebraic Multi-Grid (AMG) methods, which are especially suited for the solution of large sparse systems of algebraic equations arising from the finite element discretization of second-order elliptic boundary value problems on unstructured, fine meshes in two or three dimensions. The only information needed is recovered from the stiffness matrix. We present two types of coarsening algorithms based on the graph of the stiffness matrix. In some special cases of nested mesh refinement, we observe, that some geometrical version of the multi-grid method turns out to be a special case of our AMG algorithms. Finally, we apply our algorithms on two and three dimensional heat conduction problems in domains with complicated geometry (e.g., micro-scales), as well as to plane strain elasticity problems with jumping coefflcients.