1. Is there a simple graph with degree sequence (1,1,3,3,4,4,7)? Justify your answer. 2. Determine whether the following graph is a simple graph or multigraph, and obtain the adjac... 1. Is there a simple graph with degree sequence (1,1,3,3,4,4,7)? Justify your answer. 2. Determine whether the following graph is a simple graph or multigraph, and obtain the adjacency matrix. 3. Draw a graph with the adjacency matrix [[0, 1, 0, 1], [1, 0, 1, 0], [0, 1, 1, 1], [1, 0, 1, 1]] with respect to the ordering of the vertices a, b, c, d. Check whether the graph H displayed is bipartite or not? 4. Check whether the graphs are isomorphic or not? 5. Which of the following graphs have Euler circuits, then construct such graph if exists. 6. Which of the following graphs have Hamilton circuit or Hamilton path, and then construct such graph if exists. 7. Determine whether the given graph is planar. If so, draw it so that no edges cross. 8. Determine all spanning trees of the graph G shown in the following Figure. 9. Construct all possible spanning trees of the graph H shown in the following Figure. 10. Apply Kruskal's Algorithm to construct the minimum spanning tree (MST) from the following graph.
Understand the Problem
The questions appear to be related to concepts in discrete mathematics, specifically focused on graph theory. They ask for analysis of different graphs, including checking properties like planarity, bipartiteness, and constructing spanning trees and minimum spanning trees.
Answer
No simple graph can exist with the degree sequence \( (1, 1, 3, 3, 4, 4, 7, 9) \).
Answer for screen readers
No, there is no simple graph with the degree sequence ( (1, 1, 3, 3, 4, 4, 7, 9) ).
Steps to Solve
- Define the graph and its degree sequence
The graph is defined by its degree sequence ( (1, 1, 3, 3, 4, 4, 7, 9) ). A degree sequence lists the degree (number of edges) of each vertex.
- Check the Handshaking Lemma
According to the handshaking lemma, the sum of the degrees must be even. Calculate the sum: $$ 1 + 1 + 3 + 3 + 4 + 4 + 7 + 9 = 32 $$ Since 32 is even, this requirement is satisfied.
- Check degree constraints
For a simple graph:
- The maximum degree must be less than the number of vertices. Here, the maximum degree is 9, implying at least 10 vertices (therefore a simple graph cannot exist).
Conclusion: No simple graph exists.
- Conclude your findings
State that because the maximum degree (9) implies more vertices than provided, no simple graph aligns with this degree sequence.
No, there is no simple graph with the degree sequence ( (1, 1, 3, 3, 4, 4, 7, 9) ).
More Information
This situation demonstrates the constraints imposed by degree sequences in graph theory. Specifically, the impossibility arises from the handshaking lemma and the necessity of having sufficient vertices.
Tips
- Miscalculating the sum of the degrees.
- Confusing simple graphs with multigraphs. It's crucial to recognize that a simple graph cannot have multiple edges between the same vertices.
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