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• Author: Andrzej Krach
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## Abstract

The present paper discusses determining diagonal branches in a mine ventilation network by means of a method based on the relationship A⊗ PT(k, l) = M, which states that the nodal-branch incidence matrix A, modulo-2 multiplied by the transposed path matrix PT(k, l ) from node no. k to node no. l, yields the matrix M where all the elements in rows k and l - corresponding to the start and the end node - are 1, and where the elements in the remaining rows are 0, exclusively. If a row of the matrix M is to contain only „0” elements, the following condition has to be fulfilled: after multiplying the elements of a row of the matrix A by the elements of a column of the matrix PT(k, l), i.e. by the elements of a proper row of the matrix P(k, l ), the result row must display only „0” elements or an even number of „1” entries, as only such a number of „1” entries yields 0 when modulo-2 added - and since the rows of the matrix A correspond to the graph nodes, and the path nodes level is 2 (apart from the nodes k and l, whose level is 1), then the number of „1” elements in a row has to be 0 or 2. If, in turn, the rows k and l of the matrix M are to contain only „1” elements, the following condition has to be fulfilled: after multiplying the elements of the row k or l of the matrix A by the elements of a column of the matrix PT(k, l), the result row must display an uneven number of „1” entries, as only such a number of „1” entries yields 1 when modulo-2 added - and since the rows of the matrix A correspond to the graph nodes, and the level of the i and j path nodes is 1, then the number of „1” elements in a row has to be 1. The process of determining diagonal branches by means of this method was demonstrated using the example of a simple ventilation network with two upcast shafts and one downcast shaft.

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## Abstract  