Graph Theory

Question

Graph Theory

Anu Bhardwaj · Accepted Answer

Step 1:- To determine whether each of the following graphs is a spanning tree of G G , we first need to understand what a spanning tree is. A spanning tree of a graph G G G is a subgraph that includes all the vertices of G G , is connected, and has no cycles. Essentially, it is a minimal subset of the edges of G G  that connects all the vertices without forming any loops. For the given graph G G : Vertices: A , B , C , D , E A, B, C, D, E Edges: { ( A , B ) , ( A , D ) , ( A , E ) , ( B , C ) , ( C , D ) , ( D , E ) } \{(A, B), (A, D), (A, E), (B, C), (C, D), (D, E)\} To be a spanning tree, a subgraph must: Include all five vertices { A , B , C , D , E } \{A, B, C, D, E\} Be connected. Contain exactly 5 − 1 = 4 5 - 1 = 4  edges (since a tree with n n  vertices has n − 1 n-1  edges). Let's analyze the graph G G  to find potential spanning trees. We can remove any two edges from the given graph G G  and check if the remaining graph is connected and includes all the vertices. Here are some potential spanning trees of G G : { ( A , B ) , ( A , D ) , ( A , E ) , ( C , D ) } \{(A, B), (A, D), (A, E), (C, D)\} { ( A , B ) , ( A , D ) , ( A , E ) , ( B , C ) } \{(A, B), (A, D), (A, E), (B, C)\} { ( A , B ) , ( A , D ) , ( D , E ) , ( B , C ) } \{(A, B), (A, D), (D, E), (B, C)\} { ( A , B ) , ( C , D ) , ( D , E ) , ( A , E ) } \{(A, B), (C, D), (D, E), (A, E)\} Explanation:- Each of these subgraphs has 4 edges, connects all the vertices, and has no cycles. If you need to determine specific spanning trees based on other provided graphs, please upload those specific graphs or provide further details so I can assist you in determining if they are spanning trees of G G . Step 2:- Graph 1: Vertices: A , B , C , D , E A, B, C, D, E Edges: { ( A , D ) , ( A , E ) , ( C , B ) , ( C , D ) } \{(A, D), (A, E), (C, B), (C, D)\} This graph includes all five vertices. It has 4 edges. It is connected and has no cycles. Therefore, Graph 1 is a spanning tree. Answer for Graph 1: A. Yes, it is a spanning tree. Graph 2: Vertices: A , B , C , D , E A, B, C, D, E Edges: { ( A , B ) , ( A , D ) , ( A , E ) , ( B , C ) , ( C , D ) } \{(A, B), (A, D), (A, E), (B, C), (C, D)\} This graph includes all five vertices. It has 5 edges. It is connected but contains a cycle (because there are more edges than the required n − 1 n-1  for a tree). Therefore, Graph 2 is not a spanning tree. Answer for Graph 2: B. No, it is not a spanning tree. Graph 3: Vertices: A , B , C , D , E A, B, C, D, E Edges: { ( A , B ) , ( A , E ) , ( C , B ) , ( D , E ) } \{(A, B), (A, E), (C, B), (D, E)\} This graph includes all five vertices. It has 4 edges. It is not connected because vertex A A  and vertex D D  are isolated from each other. Therefore, Graph 3 is not a spanning tree. Graph 4: Vertices: A , B , C , D , E A, B, C, D, E Edges: { ( B , C ) , ( C , D ) , ( D , E ) } This graph does not include all five vertices (vertex A A  is missing). It has 3 edges. Therefore, Graph 4 is not a spanning tree. Answer for Graph 4: B. No, it is not a spanning tree. Graph 5: Vertices: A , B , C , D , E A, B, C, D, E Edges: { ( A , D ) , ( A , E ) , ( C , D ) , ( B , E ) } \{(A, D), (A, E), (C, D), (B, E)\} This graph includes all five vertices. It has 4 edges. It is connected and has no cycles. Therefore, Graph 5 is a spanning tree. Answer for Graph 5: A. Yes, it is a spanning tree. Graph 6: Vertices: A , B , C , D , E A, B, C, D, E Edges: { ( A , E ) , ( A , D ) , ( C , D ) } \{(A, E), (A, D), (C, D)\} This graph does not include all five vertices (vertex B B  is missing). It has 3 edges. Therefore, Graph 6 is not a spanning tree. Answer for Graph 6: B. No, it is not a spanning tree. Explanation:- Graph 1: Answer: A. Yes, it is a spanning tree. This graph includes all five vertices. It has 4 edges. It is connected and has no cycles. Graph 2: Answer : B. No, it is not a spanning tree. This graph includes all five vertices. It has 5 edges. It is connected but contains a cycle (because there are more edges than the required  𝑛 − 1  for a tree). Graph 3: Answer : B. No, it is not a spanning tree. This graph includes all five vertices. It has 4 edges. It is not connected because vertex  𝐴  and vertex  𝐷  are isolated from each other. Graph 4: Answer : B. No, it is not a spanning tree. This graph does not include all five vertices (vertex  𝐴  is missing). It has 3 edges. Therefore, Graph 4 is not a spanning tree. Graph 5: Answer: A. Yes, it is a spanning tree. This graph includes all five vertices. It has 4 edges. It is connected and has no cycles. Therefore, Graph 5 is a spanning tree. Graph 6: Answer : B. No, it is not a spanning tree. This graph does not include all five vertices (vertex  𝐵  is missing). It has 3 edges. Therefore, Graph 6 is not a spanning tree. Final Answer:- Graph 1: Answer:  A. Yes, it is a spanning tree. Graph 2: Answer :  B. No, it is not a spa

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Anu Bhardwaj

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