Problem 1 - Discrete Mathematics with Applications

Question

Determine if the simple graphs are isomorphic. When they are, determine an isomorphism \(f.\)

Step-by-Step Solution

Answer

To determine if the given simple graphs are isomorphic, first count the number of vertices and their degrees in each graph. Compare this information; if the graphs have the same number of vertices with the same degrees and the overall structure is the same, identify potential isomorphism mappings. Test the isomorphism mapping by checking if it's one-to-one and if the connectivity between vertices is preserved. If these conditions are satisfied, the graphs are isomorphic and the mapping represents the isomorphism \(f\). Otherwise, they are not isomorphic.

Step 1: Count the number of vertices and their degrees in each graph

For both graphs, count the number of vertices and the degree of each vertex. Remember that the degree of a vertex is the number of edges connected to it.

Step 2: Compare the information from Step 1

Compare the number of vertices in each graph and the degrees for each vertex. If the two graphs have the same number of vertices with the same degrees, proceed to Step 3. If not, the graphs are not isomorphic and we can stop here.

Step 3: Compare the overall structure of each graph

Examine the structure of each graph, including whether or not they have the same number of edges and any subgraphs (e.g., cycles, cliques) of the same size. If there are discrepancies in the overall structure or the properties of the graphs, they are not isomorphic and we can stop here.

Step 4: Identify potential isomorphism mappings

Attempt to map vertices in one graph to vertices in the other graph, such that vertices of the same degree are mapped to each other. Also, make sure that vertices that are connected by an edge in one graph are mapped to vertices that are connected by an edge in the other graph.

Step 5: Test the isomorphism mapping

Once you have a potential mapping for isomorphism: 1. Make sure that the mapping is one-to-one, meaning each vertex from the first graph is uniquely mapped to a vertex in the second graph and vice versa. 2. Verify that the connectivity between vertices in the mapping is preserved, meaning that if two vertices are connected by an edge in the first graph, their corresponding mapped vertices in the second graph must also be connected by an edge. 3. If the mapping satisfies both of the above conditions, then the graphs are isomorphic and the mapping represents the isomorphism f. In case we find an isomorphic mapping fulfilling these conditions, then we can conclude that the graphs are isomorphic. Otherwise, they are not isomorphic.