Graph theory plays very crucial role in understanding of complicated electrical circuits. Now what that actually means is a circuit consisting of more than six loops are very complicated to handle manually with pen and paper. If we have computer with multisim then it becomes a easy to handle task but without computer task become too complicated so to simply analysis of such circuit we often prefer Graph Theory.
How are Graphs Formed?
Each element present in circuit act as single branch of graph whereas Active sources like voltage source and current source are replace with their internal impedance (replace voltage source with short circuit whereas replacing current source with open circuit).
Oriented Graph is one which contains direction on each branch which are direction of flow of current in electric circuit to be analysed.
Planar graph is graph which can be represented on plane without crossing any other branch. Below figure show an example of graph that is planar in nature since no branch cuts any other branch in graph.
At first sight it looks as non planar graph since two resistor cross each other but it is planar graph which can be drawn as shown below.
It is not always that graph which looks non-planar is always non-planar with some modification it can be made planar. See example below for explanation.
Non planar graph is one which cannot be represented on paper without crossing other branch.