Investigating further questions around this question: [Example of sequence of graphs which satisfy the Riemann hypothesis?](https://mathoverflow.net/questions/432283/example-of-sequence-of-graphs-which-satisfy-the-riemann-hypothesis) leads to the partition function $Z$ of the Ising model of the graph defined here: [Why is this bipartite graph a partial cube, if it is?](https://mathoverflow.net/questions/432018/why-is-this-bipartite-graph-a-partial-cube-if-it-is)


It seems that:


$$Z\_{n,\beta} = \sum\_{k=0}^N c\_{N,k} \cdot \exp(-\beta)^{2(N-2k)}$$


where $N = \sum\_{i=1}^{n-1} \omega(i) = |E\_{n-1}|$ and the coefficients $c\_{N,k}$ are somehow mysterious to me at the moment.


Here are some empirical data, where $t = \exp(-\beta)$:



```
n N Z_n
1 0 2
2 0 4
3 1 4*t^2 + 4/t^2
4 2 4*t^4 + 4/t^4 + 8
5 3 4*t^6 + 12*t^2 + 12/t^2 + 4/t^6
6 4 4*t^8 + 16*t^4 + 16/t^4 + 4/t^8 + 24
7 6 4*t^12 + 8*t^8 + 28*t^4 + 28/t^4 + 8/t^8 + 4/t^12 + 48
8 7 4*t^14 + 12*t^10 + 36*t^6 + 76*t^2 + 76/t^2 + 36/t^6 + 12/t^10 + 4/t^14
9 8 4*t^16 + 16*t^12 + 48*t^8 + 112*t^4 + 112/t^4 + 48/t^8 + 16/t^12 + 4/t^16 + 152

```

**Question: What are the coefficients $c\_{N,k}$?**


I am not asking for a proof, just some empirical direction, if it is possible and not so difficult to do.


[Here](https://sagecell.sagemath.org/?z=eJyNVE2P2jAUvCPxH7z0YgeDCEckHyt2hZZLeyKKVk5jsobkJWsb2v33teMQJ6Bqm1Nsv5l5H2Pn4ohyqc0On4UCsplOkP2UMBcFqORVlnPEabZB-kOZNgZzysm8_ctoRhbrqNvNCJlOppPcMgLEdsPx0p5TVk2tDIJL1XwirhE0fr-0UZ-IocSFtzToWCuUIQlIcSgEjinMY4LkEfEnlqUd3dEjGUvSTmGQedJFVaVlriTgNpb4zd8SQChtT3CS_Vusz4cxS5N24FGDOqZb2YUwu63izXvbqFD61kr5_Y7EafJ7zUEV2yXP87erUEb8wZyEgz7Z0OEdDmNCGxSGEfTRmFfkVtOFjGrZ3oJu1ZzLumjDxqbQlwpfeXnhRtZOqSFRWGZ22SbZuCQbJSvxlsur1LXSuPiV-8x6n9SVKDh-sJ0AfAeFgHnnlSyN1bqHucQ84dnncL7vcMvhWX4qaQ9KgYGe73jAWXow0h_iI2hducIzMxu3LjFRhNcRHpGShyRWbRLpgPqAgcXUsBsrddNjrvFD64xtxby3GHQ5vNqIdRSBXx3sahVMVgX116EbtHP_CxhRCIUrssxlIY3Ga9rw3NQ9t_ueXajrbYIXMYkirROZRv3_KW2FZBACX_hptGPvE5b0RPc1COJOtktnQ41JSsK18hXMGXINXTxr8uCLHkboweOcFgwnHa9ulRqb-Xhg36yvwOCZteKMdgMAOwDjOz-6SUO7DrH2OfkS656cIfa7pu1ovoD1V66HQWyBEA-gi_g_wXsLcgR-5ZMHuqcH8hcugabu&lang=sage&interacts=eJyLjgUAARUAuQ==) is the python source code for further computations, if needed.


If it is of interest to the question:


The Hamiltonian function is maximal possible:


$$H\_n = - \sum\_{(u,v) \in E\_n} \sigma\_u \sigma\_v 
 = |E\_n| = \sum\_{k=1}^n \omega(n) \approx n(\log(\log(n))+B\_1)+o(n)$$


where $B\_1 \approx 0.26$ is Mertens constant.



What are the coefficients of this partition function in the following Ising model?
In general, the partition function of the Ising model is usually nicer when defining $x = \tanh(\beta)$ instead of $t = \exp(- \beta)$. One explanation for this is that the Ising model partition function is the same as the partition function for the random cluster model and the partition function of the Ising model can be calculated using the loop-O(1)-model with parameter $x = \tanh(\beta)$ and Bernoulli percolation with parameter $x$.


I tried to adopt this point of view here: <https://alea.impa.br/articles/v19/19-07.pdf>


Let me give just an example with the graph which is just a loop of $n$ vertices and $n$ edges. Then I can find the partition function by taking all even (spanning) subgraphs of the graph (which means subsets of edges such that all vertices have even degree.) In this particular case it is the full subgraph and the empty one. Then the partition function of the Ising model is given as


$$
Z = \sum\_{g \text{ even subgraph of } G} x^{o(g)} 
$$
where $o(g)$ is the number of open edges of $g$ meaning that $o(\emptyset) = 0$ and $o(G) = n$ in the particular case.


Thus, we have that $ Z = 1 + x^n$ in that particular case.
I do not really understand the definition of your graphs, but this point of view might be helpful.


