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

[vs140580]

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.

 

[HOI]

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?

 

[vs140580]

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.