How many graph isomorphic classes are there given n vertices?

Thread starter Dragonfall
Start date Jun 11, 2009
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In summary, graph isomorphism is a concept in graph theory that refers to the rearrangement of vertices and edges in a graph to create an identical graph. The number of isomorphic classes can be calculated using the Polya enumeration theorem, and there are efficient algorithms for determining this number. The number of isomorphic classes can vary depending on the characteristics of the graph. Graph isomorphism is used in various real-world applications, such as chemistry, computer science, and social network analysis.

Jun 11, 2009

Dragonfall

I'm talking about undirected simple graphs.

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Jun 12, 2009

CRGreathouse

Science Advisor

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I believe this is
http://www.research.att.com/~njas/sequences/A000088

Last edited by a moderator: May 4, 2017

Jun 12, 2009

Dragonfall

Thanks. I'd like to see how those formulas were obtained, though.

Jun 12, 2009

maze

Brute force??

Jun 12, 2009

Tac-Tics

The formula is at the bottom of the link above.

Jun 12, 2009

Dragonfall

Yes but how do you derive said formula?

1. How do you define graph isomorphism?

Graph isomorphism is a concept in graph theory that refers to the ability to rearrange the vertices and edges of a graph to create an identical graph. In other words, two graphs are isomorphic if they have the same number of vertices, the same number of edges, and the same degree sequence.

2. How is the number of graph isomorphic classes calculated given n vertices?

The number of graph isomorphic classes for a given number of vertices can be calculated using the Polya enumeration theorem. This theorem involves counting the number of ways to color the vertices of a graph with a certain number of colors, while taking into account the symmetries of the graph. The result is then divided by the number of symmetries to obtain the number of isomorphic classes.

3. Are there any efficient algorithms for determining the number of graph isomorphic classes?

Yes, there are several efficient algorithms for determining the number of graph isomorphic classes. Some examples include the Weisfeiler-Lehman algorithm, which uses a labeling approach, and the color refinement algorithm, which iteratively refines the coloring of a graph until it reaches a stable state.

4. Is the number of graph isomorphic classes the same for all possible graphs with n vertices?

No, the number of graph isomorphic classes can vary depending on the specific characteristics of the graph, such as the degree sequence and symmetries. For example, a complete graph and a cycle graph with the same number of vertices will have a different number of isomorphic classes.

5. How is the concept of graph isomorphism used in real-world applications?

Graph isomorphism has many practical applications, such as in chemistry for determining the structural similarity of molecules and in computer science for optimizing data structure representations. It is also used in social network analysis to identify similar patterns of connections between individuals.