Consider the following circuits :
The planner circuits are
The circuit 1 and 2 are redrawn as below. 3 and 4 can not be redrawn on a plane without crossing other branch.
Consider the following graphs
Nonplanner graphs are
Other three circuits can be drawn on plane without crossing
A graph of an electrical network has 4 nodes and 7 branches. The number of links l, with respect to the chosen tree, would be
l = b  (n  1) = 4.
For the graph shown in fig.correct set is
There are 4 node and 6 branches.
t = n  1 = 3, l = b  n + 1 = 3
A tree of the graph shown in fig. is
From fig. it can be seen that a f h g is a tree of given graph
Branch current and loop current relation are expressed in matrix form as
where i_{j} represent branch current and I_{k} loop current. The number of independent node equation are
Number of branch = 8
Number of link = 4
Number of twigs = 8  4 = 4
Number of twigs = number of independent node equation.
If the number of branch in a network is b, thenumber of nodes is n and the number of dependent loop is l, then the number of independent node equations will be
The number of independent node equation are n  1.
Branch current and loop current relation are expressed in matrix form as
.
where i_{j} represent branch current and I_{k} loop current.
The rank of incidence matrix is
Number of branch b = 8
Number of link l = 4
Number of twigs t = b  l = 4
rank of matrix = n  1 = t = 4
A network has 8 nodes and 5 independent loops. The number of branches in the network is
Independent loops = link
l = b  (n  1)
= 5 = b  7
b = 12
A branch has 6 node and 9 branch. The independent loops are
Independent loop = link
l = b  (n1) = 4
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