91Ó°ÊÓ

There is only one simple graph with 0 edges: four disconnected vertices, labeled A, B, C, and D. We'll call this graph G0.

There's only one simple graph with 1 edge: connect any two vertices (e.g., A to B) and leave the other two disconnected. We'll call this graph G1.

For two edges, we have two possibilities: 1. Connecting two pairs of vertices (A-B and C-D): Call this graph G2. 2. Connecting three vertices in a line or a path (A-B, B-C): Call this graph G3.

For three edges, we have two possibilities: 1. Connecting all three vertices to one central vertex (A-B, A-C, and A-D): Call this graph G4. 2. Forming a triangle and connect the fourth vertex to one vertex (A-B, B-C, C-A, A-D): Call this graph G5.

There's only one simple graph with 4 edges: forming a cycle with all four vertices (A-B, B-C, C-D, D-A). Call this graph G6.

For a simple graph with 4 vertices and 5 edges, we'd need to connect every vertex to every other vertex (A-B, B-C, C-D, D-A, A-C). Call this graph G7.

We need to compare the drawn graphs and check if any of them are isomorphic. If yes, we remove one of the isomorphic graphs from our list. In this case, none of the drawn graphs are isomorphic, so we have the following nonisomorphic simple graphs with four vertices: 1. G0: No edges (4 disconnected vertices) 2. G1: One edge (two vertices connected and two disconnected vertices) 3. G2: Two edges forming two disconnected pairs of connected vertices 4. G3: Two edges forming a path of three vertices and one disconnected vertex 5. G4: Three edges connecting all vertices to a central vertex 6. G5: Four edges forming a triangle and an additional edge from one of the triangle vertices to the fourth vertex 7. G6: Four edges forming a cycle with all four vertices 8. G7: Five edges connecting every vertex to every other vertex (a complete graph) These are all the nonisomorphic simple graphs with four vertices.