What is this equation for coin operator is trying to do in this quantum walk for Non-regular graph? This coin operator is called Fourier coin

Question

I am reading the following paper: Discrete-time quantum walk on complex networks for community detection by Kanae Mukai
We define the Coin operator $C$ by: $C=C_1\otimes C_2....C_n$ , We define coin operator for Node $i, C_i:H_i\to H_i$ is given by:

$C_i^F|i\to j_1\rangle|i\to j_2\rangle.......|i\to j_k\rangle=(|i\to j_1\rangle|i\to j_2\rangle.......|i \to j_k\rangle)1/\sqrt(k_i) \begin{pmatrix} 1 & 1 & 1 & ... & 1\\ 1 & e^{\iota\theta/k_i} & e^{2\iota\theta/k_i} & ... & e^{(k_i-1)\iota\theta/k_i}\\ 1 & e^{2\iota\theta/k_i} & e^{4\iota\theta/k_i} & ... & e^{2(k_i-1)\iota\theta/k_i}\\ . & . & . & . & .\\. & . & . & . & .\\. & . & . & . & .\\ 1 & e^{(k_i-1)\iota\theta/k_i} & e^{2(k_i-1)\iota\theta/k_i} & ... & e^{(k_i-1)(k_i-1)\iota\theta/k_i}\end{pmatrix}$
Here $k_i$ is the degree of $i^{th}$ node and $\theta=2\pi$. The author called this coin as Fourier Coin. And this $i\to j$ implies that Node $i$ is going to hope to adjacent Node $j$. Now, What is going on with this equation?

Answer 1

The operator Fourier Coin is $k$-point Discrete Fourier Transform (DFT) of node $i$. The matrix representation of a general $N$-point DFT can be found here.

The implementation of DFT on the quantum computer is what we know as QFT, and it can be found in Mike and Ike on pg.219. More specifically, an eight-point DFT can be implemented on the quantum computer as