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Consider the following statements regarding trees: 1. A tree contains all the nodes of the graph. 2. A tree shall contain any one of the loops. 3. Every connected graph has at least one tree. Which of the above statements are correct?

Consider the following statements regarding trees: 1. A tree contains all the nodes of the graph. 2. A tree shall contain any one of the loops. 3. Every connected graph has at least one tree. Which of the above statements are correct? Correct Answer 1 and 3 only

Let us consider the following electric circuit:

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An equivalent graph corresponding to the above electric circuit is shown in the following figure

:

[ alt="F1 S.B 29.8.20 Pallavi D5" src="//storage.googleapis.com/tb-img/production/20/08/F1_S.B_29.8.20_Pallavi_D5.png" style="width: 174px; height: 177px;">

Tree:

The tree is a connected subgraph of a given graph, which contains all the nodes of a graph. But, there should not be any loop in that subgraph. The branches of a tree are called as twigs.

[ alt="F1 S.B 29.8.20 Pallavi D6" src="//storage.googleapis.com/tb-img/production/20/08/F1_S.B_29.8.20_Pallavi_D6.png" style="width: 173px; height: 127px;">

 

This connected subgraph contains all the four nodes of the given graph and there is no loop.

Hence, it is a Tree.

Procedure

  • In a tree, there exists one and only one path between any pair of nodes.
  • Every connected graph has at least one tree.
  • Every tree has (n – 1) branches, where n is the number of nodes of the tree.
  • The rank of a tree is n – 1.
  • A tree contains all the nodes of a graph.

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Graph:

The network graph is simply called a graph.

  • It consists of a set of nodes connected by branches. In graphs, a node is a common point of two or more branches. Sometimes, only a single branch may connect to the node.
  • A branch is a line segment that connects two nodes.
  • Any electric circuit or network can be converted into its equivalent graph by replacing the passive elements and voltage sources with short circuits and the current sources with open circuits.

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