How Many Unique Substructures Exist in an n-Clique?

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In summary, the n-clique K_n has 2^n substructures, n substructures up to isomorphism, and 2^n elementary substructures.
  • #1
Andrei1
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Let \(\displaystyle K_n\) be the \(\displaystyle n\)-clique for some \(\displaystyle n\in\mathbb{N}\). Then any graph having at most \(\displaystyle n\) vertices is a subgraph of \(\displaystyle K_n\).
(a) How many substructures does \(\displaystyle K_n\) have?
(b) How many substructures does \(\displaystyle K_n\) have up to isomorphism?
(c) How many elementary substructures does \(\displaystyle K_n\) have?
My answers:
(a) \(\displaystyle 2^n\)
(b) \(\displaystyle n\)
(c) \(\displaystyle 2^n\)
Are they correct?
 
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  • #2
Andrei said:
My answers:
(a) \(\displaystyle 2^n\)
(b) \(\displaystyle n\)
(c) \(\displaystyle 2^n\)
Are they correct?
If by substructures you mean subgraph, then the answer to part (a) looks okay. Some people may write $2^n-1$ because they might not want to include $\emptyset$ in the list.

For the second, is the question asking to find out the number of graphs, up to isomorphism, having at most $n$ vertices. In that case, the answer cannot be just $n$.

For part (c), I don't know the definition of elementary substructures. Can you please define it?
 
  • #3
\(\displaystyle M\) is a substructure of \(\displaystyle N\) ... if
1. \(\displaystyle M\) is a structure having the same vocabulary as \(\displaystyle N\),
2. the underlying set \(\displaystyle U_M\) of \(\displaystyle M\) is a subset of the underlying set \(\displaystyle U_N\) of \(\displaystyle N\), and
3. \(\displaystyle M\) interprets the vocabulary in the same manner as \(\displaystyle N\) on \(\displaystyle U_M\).

Let \(\displaystyle N\) and \(\displaystyle M\) be structures in the same vocabulary. Then \(\displaystyle M\) is an elementary substructure of \(\displaystyle N\) ... if and only if the identity function \(\displaystyle id\colon M\to N\) defined by \(\displaystyle id(x)=x\) is an elementary embedding [preserves all formulas].

About (b). Let's take a 4-clique. Any subgraph of it is not a substructure if it is not a clique. So I have, up to isomorphism, 4 substrucutures: one 1-clique, one 2-clique, ...
 
  • #4
Andrei said:
About (b). Let's take a 4-clique. Any subgraph of it is not a substructure if it is not a clique. So I have, up to isomorphism, 4 substrucutures: one 1-clique, one 2-clique, ...
Then by substructures you do not means subgraphs. It's fine then.
 

FAQ: How Many Unique Substructures Exist in an n-Clique?

What is an n-clique?

An n-clique is a subgroup within a larger group or network, where each member is directly connected to every other member in the subgroup.

How are substructures of the n-clique determined?

Substructures of the n-clique are determined by examining the connections between members within the larger group. If all members within a subgroup are directly connected to each other, then it is considered a substructure of the n-clique.

What is the significance of studying substructures of the n-clique?

Studying substructures of the n-clique can provide insights into the structure and organization of a larger group or network. It can also help identify key members or nodes within the group and their role in maintaining connectivity.

Can substructures of the n-clique be found in real-world networks?

Yes, substructures of the n-clique can be found in various real-world networks such as social networks, transportation networks, and biological networks.

Are there any applications for understanding substructures of the n-clique?

Understanding substructures of the n-clique can have practical applications in fields such as social network analysis, epidemiology, and transportation planning. It can also aid in identifying key players or influencers within a network.

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