Math  /  Discrete

QuestionConsider the network shown to the right. (a) Find a spanning tree of the network. (b) Calculate the redundancy of the network. (c) What is the diameter of the network?

Studdy Solution

STEP 1

1. A spanning tree is a subgraph that includes all the vertices of the original graph and is a single connected tree.
2. Redundancy in a network graph is calculated as the difference between the number of edges in the graph and the number of edges in a spanning tree.
3. The diameter of a network is the longest shortest path between any two nodes in the graph.

STEP 2

1. Identify a spanning tree for the network.
2. Calculate the redundancy of the network.
3. Determine the diameter of the network.

STEP 3

To find a spanning tree, we need to select edges that connect all nodes without forming any cycles.
One possible spanning tree can be constructed by choosing the following edges: - A to B - B to C - C to D - D to E
This selection connects all nodes: A, B, C, D, and E, without forming any cycles.

STEP 4

Calculate the redundancy of the network.
First, count the total number of edges in the original graph: - A to B - A to E - A to D - B to C - B to D - C to D - D to E
Total edges = 7
The number of edges in the spanning tree is 4 (from STEP_1).
Redundancy = Total edges - Edges in spanning tree = 7 - 4 = 3

STEP 5

Determine the diameter of the network.
The diameter is the longest shortest path between any two nodes. In the spanning tree found in STEP_1, the longest path is: - A to B to C to D to E
This path has 4 edges.
Thus, the diameter of the network is 4.
The results are: (a) A possible spanning tree includes edges: A-B, B-C, C-D, D-E. (b) The redundancy of the network is 3. (c) The diameter of the network is 4.

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