Topological matrices. Incidence matrices

2.2 Topological matrices

For an analytical description of electric circuit graphs and their storage in computer memory in a digital form it is more convenient to represent graphs in the form of topological matrices. There are incidence matrices (node matrices), loop matrices and graph section matrices.

2.2.1 Incidence matrices

It is said that if the node is the end of the branch then they are incident. Information contained in a directed graph can be fully represented by a matrix called incidence matrix (node matrix).

The matrix is called the incidence matrix which corresponds to the directed graph with “ ” nodes and “ ” branches

(2.1)

where is the element of the matrix;

=1 if the branch is incident to the node and directed from the node; =-1 if the branch is incident to the node and directed to the node; =0 if the branch is not incident to the node . For instance, we will obtain the matrix (2.2) for the directed graph according to Fig. 2.3,c. It’s clear from the matrix that the number of non – zero elements in each line of the matrix is equal to the number of branches incident to the corresponding node. Each column contains only

branches

= (2.2)

incident branches

two non – zero elements: “+1” and “-1” because each branch is incident to two nodes and directed from one of them to the other. The sum of all elements of each column and, consequently, the sum of all matrix lines is equal to zero, i.e. the matrix lines are linearly dependent. Therefore, it’s possible to exclude any line of the matrix without any information loss. So, when the 4-th line is excluded in (2.2) we get:

= (2.3)

Matrix is called the reduced incidence matrix.

⇐ Предыдущая 123 4 Следующая ⇒