We consider the block matrices and 3-dimensional graph manifolds associated with a special type of tree graphs. We demonstrate that the linking matrices of these graph manifolds coincide with the reduced matrices obtained from the Laplacian block matrices by means of Gauss partial diagonalization procedure described explicitly by W. Neumann. The linking matrix is an important topological invariant of a graph manifold which is possible to interpret as a matrix of coupling constants of gauge interaction in Kaluza-Klein approach, where 3-dimensional graph manifold plays the role of internal space in topological 7-dimensional BF theory. The Gauss-Neumann method gives us a simple algorithm to calculate the linking matrices of graph manifolds and thus the coupling constants matrices.

We present a new algorithm for the fast expansion of rational numbers into continued fractions. This algorithm permits to compute the complete set of integer Euler numbers of the sophisticate tree graph manifolds, which we used to simulate the coupling constant hierarchy for the universe with five fundamental interactions. Moreover, we can explicitly compute the integer Laplacian block matrix associated with any tree plumbing graph. This matrix coincides up to sign with the integer linking matrix (the main topological invariant) of the graph manifold corresponding to the plumbing graph. The need for a special algorithm appeared during computations of these topological invariants of complicated graph manifolds since there emerged a set of special rational numbers (fractions) with huge numerators and denominators; for these rational numbers, the ordinary methods of expansion in continued fraction became unusable.

The hierarchy and fine tuning of the gauge coupling constants are described on the base of topological invariants (Chern classes interpreted as filling factors) characterizing a collection of fractional topological fluids emerging from three dimensional graph manifolds, which play the role of internal spaces in the Kaluza-Klein approach to the topological BF theory. The hierarchy of BF gauge coupling constants is simulated by diagonal elements and eigenvalues of rational linking matrices of tree graph manifolds pasted together from Brieskorn (Seifert fibered) homology spheres. Specific examples of graph manifolds are presented which contain in their linking matrices the hierarchy of coupling constants distinctive for the dimensionless coupling constants in our Universe. The fine tuning effect is simulated owing to the special numerical properties of diagonal elements of the linking matrices. We pay a particular attention to fine tuning problem for the cosmological constant.