Abstract
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.
Highlights
Graphs can serve as a universal remedy for the codification and classification of topological spaces, matrices, dynamical systems, etc
We consider the following question: how the topological invariants of manifold corresponding to a tree graph 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 [1]
Summary
Graphs can serve as a universal remedy for the codification and classification of topological spaces, matrices, dynamical systems, etc. 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 [1]. We conclude formulating our main results and considering an example of their application for the topological field theory
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