Graph theory has many applications in several fields such as engineering, physical, social and biological sciences, linguistics etc. Any physical situation that involves discrete objects with interrelationships can be represented by a graph. In Electrical Engineering Graph Theory is used to predict the behavior of the network in analysis. However, for smaller networks node or mesh analysis is more convenient than the use of graph theory. It may be mentioned that Kirchhoff was the first to develop theory of trees for applications to electrical network. The advent of high speed digital computers has made it possible to use graph theory advantageously for larger network analysis.
ELEMENT OF A GRAPH:
Each network element is replaced by a line segment or an arc while constructing a graph for a network. Each line segment or arc is called an element. Each Potential source is replaced by a short circuit. Each current source is replaced by an open circuit.
NODE OR VERTEX:
The terminal of an element is called a node or a vertex.
EDGE:
An element of a graph is called an edge.
DEGREE:
The number of edges connected to a vertex or node is called its degree.
GRAPH:
An element is said to be incident on a node, if the node is a terminal of the element. Nodes can be incident to one or more elements. The network can thus be represented by an interconnection of elements. The actual interconnections of the elements gives a graph.
RANK:
The rank of a graph is n - I where n is the number of nodes in the graph.
SUB GRAPH:
Any subset of elements of the graph is called a sub-graph. A sub-graph is said to be proper if it consists of strictly less than all the elements and nodes of the graph.
PATH:
A path is defined as a sub-graph of connected elements Such that not more than two elements are connected to anyone node. If there is a path between every pair of nodes then the graph is said to be connected. Alternatively, a graph is said to be connected if there exists at least one path between every pair of nodes.
PLANAR GRAPH:
A graph is said to be planar, if it can be drawn without-out cross-over of edges. Otherwise it is called non-planar. As shown in figure 1.
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Figure 1 (a) Planar Graph (b) Non-Planar Graph
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CLOSED PATH OR LOOP:
The set of elements traversed starting from one node and returning to the same node form a closed path or loop.
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Figure 2 (a) Power system single-line diagram (b) Positive
sequence network diagram
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ORIENTED GRAPH:
An oriented graph is a graph with direction marked for each element Figure 2(a) shows the single line diagram of a simple power network consisting of Generating stations, Transmission lines and loads. Figure 2(b) shows the positive sequence network of the system in Figure 2(a). The oriented connected graph is shown in Figure 3 for the same system.
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Figure 3 Oriented connected graph.
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