The section matrix

2.2.3 The section matrix

Topological matrixes can be compiled for graph sections too.

The section matrix , corresponding to the directed graph with “ ” sections and “ ” branches is the name for the matrix .

= (2.11)

where is an element of the matrix , =1, if the branch is incident to the section and coincides with the direction of the section; =-1, if the branch is incident to the section and opposite to the direction of the section; =0, if the branch is not incident to the section . For instance, for the directed graph in Fig. 2.2,c it is possible to choose 7 sections: , , , , ,

Section directions are marked with arrows on section lines. As a result, we get a section matrix

branches

= (2.12)

sections

It is obvious, that some sections used in the matrix (2.12) are linearly dependent. Only a section that includes at least one branch not being part of any other section is considered to be linearly independent. The number of independent sections, evidently, is equal to the number of independent nodes. Therefore, it is possible to retain any three lines in the matrix (2.12) without losing any information. So, having excluded the first four lines, we get the matrix:

= (2.13)

The matrix is called the base section matrix. The use of the tree pre-supposes use the systematic method of construction of the base section matrix, that is, the base matrix is to correspond to the graph main sections. Such a matrix is called the main section matrix . Directions of main sections are taken according to the direction of the corresponding edges of the graph tree. So, having arranged the edges in the lower columns we get the matrix :