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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.

Graphs can serve as a universal remedy for the codification and classification of topological spaces, matrices, dynamical systems, etc. In this article, we consider the following question: how the topological invariants of manifold corresponding to a tree graph (graph manifold) can be calculated using the method of Gauss-Neumann partial diagonalization of Laplacian matrix defined for the same graph. This calculation can be useful in multidimensional models of Kaluza-Klein type, where coupling constants of gauge interactions are simulated by the rational linking matrices of the internal space [

The paper is organized as follows. In section 2 we define the type of tree graphs considered in this paper (plumbing graphs) and Laplacian matrices for these plumbing graphs. We recall also the Gauss-Neumann method of partial diagonalization by means of which we obtain the reduced rational tridiagonal matrix for each plumbing graph. In section 3 we construct graph manifolds codified by the plumbing graphs defined in section 2 and calculate the main topological invariant for these 3-dimensional manifolds, namely rational linking matrix. Then 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-Neumann partial diagonalization procedure. Finally, we conclude formulating our main results and considering an example of their application for the topological field theory.

We begin from the definition of graph

In this paper we shall considered only the simplest type of graphs which are called plumbing graphs. An example of plumbing graph ^{I} with

A maximal internal chain of length k

A maximal terminal chain of length k

The plumbing graph

to each node a weight equal to zero that is connected with using of the unnormalized Seifert invariants for Bh-spheres, which are the block elements for the construction of graph manifolds [

Now let’s define a Laplacian matrix for the plumbing graph

with integer numbers

Note that

and notice that using Gauss-Neumann partial diagonalization [

where

and

Applying the general Gauss-Neumann partial diagonalization method for the matrix

where

chain. We have used the notation

verse, i.e.

which is represented as a continued fraction, and thus reduce the original block tridiagonal matrix

In this section we will construct a plumbing graph manifold ^{1}-(U(1)-)bundle, corresponding to i-th vertex^{1}-bundle over

where

Note that the above is a well known description of the lens space^{1}-bundles, we must use the trivial bundles over annuli

to a boundary component

For example, the plumbing of the chain shown in

where

Now recall that each edge

This set of tori performs the well known JSJ-decomposition of the graph manifold

By construction, each piece

Note that there exists an uncertainty in the choice of the torus

Suppose that we perform the plumbing operation according to the plumbing diagram

We construct the plumbing graph

A plumbing (splicing) diagram Δ_{p}

where

For internal chains the integer Euler numbers

where the Seifert (orbital) invariants

for the ordering fixed by the plumbing diagram in

where

For the cases

Moreover ^{I} and

in the following sense. Recall that edges of ^{I} and

Subindices 2 and 3 manifest that

where

Now we introduce the one-form bases

where the integrals are calculated over any such section line or fiber as, for example, in [

We suppose that the forms

Also we shall used the integrals

which define the linking (intersection) numbers of the fiber structures

The rational numbers

Now we are ready to calculate the rational linking matrix for the graph manifold

We integrate here over the three dimensional graph manifold

From the tree structure of the graph

for

If

Here we use the decomposition (6) of the piece

also known as the Chern class of line V-bundle associated with the Seifert fibration of

For

Here we have used the decompositions (8) and (9) as well as the notations (4).

Comparing the reduced matrix

We want to conclude with an example of an application of our results for the topological field theory. In [

whose elements are all rational and the diagonal ones are described in (14) and (17) by a sum of three continued fractions. The matrix

In the 7-dimensional Kaluza-Klein approach to the topological field theory (BF-model), the rational linking matrices of the 3-dimensional graph manifold may be really interpreted as coupling constants matrices [