<p>I’m a tensor-network newbie and am working through Bridgeman and Chubb’s “Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks” (arXiv:1603.03039v4).</p>
<p>At the end of Chapter 1, Problem #3 says “Calculate the contraction of the tensor network in Eq. (1.19) for bond dimension 3 …” I thought the dimension in which a contraction between two tensors would happen is determined by the set of overlapping indices. What does “contracting for bond dimension x” mean for tensor networks?</p>

CC BY-SA 4.0
2023-01-25T22:38:48.977Z
In this particular case, I think they're asking you to do the contraction for the specified tensor network when all the bonds have dimension 3 (which would indeed corresponding to counting 3-colorings, as they mention).
Joshua Grochow
CC BY-SA 4.0
2023-01-26T22:53:14.500Z
Is it more accurate to say the bonds have maximum dimension 3?  Thank you!
spreinhardt
CC BY-SA 4.0
2023-01-27T00:05:03.657Z
In this particular case, I believe they want all the bonds to have dimension exactly 3. The idea is: the 3 choices coming out of a vertex in the original graph (e-tensor in the tensor network) correspond to the 3 choices of color. The e-tensor enforces that all the colors coming out of the same vertex are the same. The n-tensors then enforce that the color of one vertex in the original graph differs from the color of its neighboring vertices.
Joshua Grochow
<tensor-rank>
What does doing a tensor network contraction "for bond dimension k" mean?
spreinhardt
# What does doing a tensor network contraction "for bond dimension k" mean?

(Asked by: username_0 on 2023-01-25)

I’m a tensor-network newbie and am working through Bridgeman and Chubb’s “Hand-waving and Interpretive Dance: An Introductory Course on Tensor Networks” (arXiv:1603.03039v4).

At the end of Chapter 1, Problem #3 says “Calculate the contraction of the tensor network in Eq. (1.19) for bond dimension 3 …” I thought the dimension in which a contraction between two tensors would happen is determined by the set of overlapping indices. What does “contracting for bond dimension x” mean for tensor networks?

**Tags:** tensor-rank

> **Comments:**
> **username_1**: In this particular case, I think they're asking you to do the contraction for the specified tensor network when all the bonds have dimension 3 (which would indeed corresponding to counting 3-colorings, as they mention).  
> **username_0**: Is it more accurate to say the bonds have maximum dimension 3?  Thank you!  
> **username_1**: In this particular case, I believe they want all the bonds to have dimension exactly 3. The idea is: the 3 choices coming out of a vertex in the original graph (e-tensor in the tensor network) correspond to the 3 choices of color. The e-tensor enforces that all the colors coming out of the same vertex are the same. The n-tensors then enforce that the color of one vertex in the original graph differs from the color of its neighboring vertices.  

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## 0 answers

