MHB Can Vector Space $(V,O_1,O_2)$ Represent 2 Graphs?

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The discussion centers on whether a vector space $(V,O_1,O_2)$ can represent two different non-isomorphic graphs. Participants explore the implications of having different operations (O_1, O_2, a_1, a_2) on the same set V and whether two distinct vector spaces can share the same basis. Clarification is sought on the nature of these operations, particularly if they can deviate from standard definitions of vector addition and scalar multiplication. The conversation emphasizes the need for a deeper understanding of vector space properties in relation to graph theory. Overall, the thread aims to enhance knowledge on the relationship between vector spaces and graph representations.
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Given a basis of a vector space $(V,O_1,O_2)$ can it represent two different non-isomorphic graphs.Any other inputs kind help. It will improve my knowledge way of my thinking.

Another kind help with this question is suppose (V,O_1,O_2) and (V,a_1,a_2) are two different vector spaces on the same set V can they both have same bases. (If not kind help with a proof or link of it to improve my knowledge.)

Here O_1,O_2,a_1,a_2 are operations on V.
 
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Here O_1,O_2,a_1,a_2 are operations on V.

By "operations", do you mean that O_1 and a_1, say, are "scalar multiplication" and O_2 and a_2 are vector addition?
 
Country Boy said:
By "operations", do you mean that O_1 and a_1, say, are "scalar multiplication" and O_2 and a_2 are vector addition?
I mean the same same but the scalar multiplication and vector addition may yes but they be defined anyway may not always be the usual way.

O_1 is vector addition and O_2 scalar multiplication of first.a_1 is vector addition and a_2 scalar multiplication of second.

On the same set V.
 

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