Non-isomorphic graphs on 3 vertices are graphs that are structurally distinct, meaning they cannot be transformed into each other simply by renaming vertices. Here is the classification of all non-isomorphic graphs on 3 vertices:

Total Possibilities

With 3 vertices, there are $2^{\binom{3}{2}} = 2^3 = 8$ possible graphs because each of the 3 edges can either exist or not exist. However, many of these graphs are isomorphic to each other. Below are the non-isomorphic ones:

1. Empty Graph

• Description: No edges are present.
• Edges: 0
• Adjacency Matrix: 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$

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2. Single Edge

• Description: A graph with one edge connecting two vertices.
• Edges: 1
• Adjacency Matrix: 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}$$

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3. Path Graph

• Description: A graph with two edges forming a "path" (no cycles).
• Edges: 2
• Adjacency Matrix: 0 & 1 & 0 \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{bmatrix}$$

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4. Complete Graph

• Description: A graph where all three vertices are connected to each other.
• Edges: 3
• Adjacency Matrix: 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{bmatrix}$$

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5. Star Graph

• Description: A graph with two edges, where both edges connect a single central vertex to two other vertices.
• Edges: 2
• Adjacency Matrix: 0 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{bmatrix}$$

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These 5 graphs are the only non-isomorphic graphs on 3 vertices.

Would you like details on any specific graph, adjacency matrices, or an explanation of isomorphism in graphs?

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Related Questions:

1. How many non-isomorphic graphs are there on 4 vertices?
2. What are the properties of each of these 3-vertex graphs?
3. How do adjacency matrices change for isomorphic graphs?
4. What is the significance of graph isomorphism in applications?
5. Can you explain how the counting of graphs scales with the number of vertices?

Tip:

When analyzing isomorphism, focus on vertex degrees, adjacency relationships, and symmetry in the graph. These properties help to quickly classify graphs!