I am currently trying to understand some results of this paper. Let's consider the first example they work through in part A, the Kagome lattice. I have changed some of the notation from the paper.
We start with a Hamiltonian of the form $$H = -t \sum_{ij} c_i^\dagger A_{ij} c_j$$ where $A_{ij}$ is the adjacency matrix of the Kagome lattice. The Kagome lattice has three sublattices, and therefore after Fourier transforming, we can write $H$ as $$H = -t \sum_{\mathbf{q}} \sum_{\mu,\,\nu} c_\mu^\dagger(\mathbf{q}) A_{\mu\nu}(\mathbf{q}) c_\nu(\mathbf{q})$$ where $\mu,\nu = 0,1,2$. Utilizing the definitions given in the paper,
along with the choice of sublattice vectors $\mathbf{c}_\mu = \left\{\mathbf{0},\mathbf{a}_1/2, -\mathbf{a}_3/2\right\}$, the resulting adjacency matrix is $$ A_{\mu\nu}(\mathbf{q}) = \left( \begin{array}{ccc} 0 & 2 \cos \left(\frac{\mathbf{q}\cdot\mathbf{a}_1}{2}\right) & 2 \cos \left(\frac{\mathbf{q}\cdot\mathbf{a}_3}{2}\right) \\ \cos \left(\frac{\mathbf{q}\cdot\mathbf{a}_1}{2}\right) & 0 & 2 \cos \left(\frac{\mathbf{q}\cdot\mathbf{a}_2}{2}\right) \\ \cos \left(\frac{\mathbf{q}\cdot\mathbf{a}_3}{2}\right) & \cos \left(\frac{\mathbf{q}\cdot\mathbf{a}_2}{2}\right) & 0 \\ \end{array} \right) $$ This matrix has three eigenvalues, $$\epsilon_0 = -2$$ $$\epsilon_\pm = 1 \pm \sqrt{3 + \Theta(\mathbf{q)}} \tag{1}$$ where $$\Theta(\mathbf{q}) = \sum_{\pm\mathbf{a}_\mu} \sin(\mathbf{q}\cdot\mathbf{a}_\mu)\tag{2}$$ Note that in the paper they defined $\Lambda$ such that $\Theta = 2\Lambda$ and restrict the sum over vectors which are not related by an overall sign.
Finally, through a unitary transformation, we can change to the eigenbasis of $A$, $A = UDU^\dagger$ where $D$ is the diagonal form of $A$ and the columns of $U$ are the normalized eigenvectors of $A$. Thus we obtain $$H = -t \sum_\mathbf{q} \sum_{\alpha\in\left\{0,\pm\right\}} \epsilon_\alpha(\mathbf{q}) a_\alpha^\dagger(\mathbf{q}) a_\alpha (\mathbf{q})$$ where $a_\alpha = \sum_\mu \left(\vec{a}_\alpha\right)^*_\mu c_\mu$ where $\vec{a}_\alpha$ is the normalized eigenvector of $A$ with eigenvalue $\epsilon_\alpha$. In particular,
$$\left(\vec{a}_0\right)_\mu \propto \sin(\mathbf{q}\cdot\mathbf{a}_{\mu+2}/2) \tag{3}$$ where the addition is modulo 3 and the $\mathbf{a}_i$ are the three shortest Bravais lattice vectors not related by an overall sign as defined in the paper.
Questions:
(A) How can we see that the eigenvalues should have the form given by (1) and (2)? I partly understand that the $\mathbf{q}$-independent eigenvalue arises from the fact that the Kagome lattice is the line graph of the honeycomb lattice which is bipartite, and the two $\mathbf{q}$-dependent eigenvalues are inherited unchanged from the honeycomb lattice (this is related to the lattice laplacian I believe). I suspect the 3 under the square root reflects the number of sublattices, and I verified that the functional form is the same for the Pyrochlore lattice (line graph of the diamond lattice) where (2) is replaced by a sum over the nearest neighbor FCC Bravais lattice vectors and the 3 is replaced by a 4 (four sublattices). The 1 changes to a 2 and I don't see an obvious connection with the geometry there, but in any case, there must be a simpler way to understand the connection between the eigenvalues and the geometry of the lattice.
(B) Why does (3) take such a seemingly simple functional form? I have no intuition for how the eigenvectors relate to the geometry here but it seems there is a deeper connection. Furthermore, why in particular is it that the $\mathbf{q}$-independent eigenvalue has a simpler eigenvector, while the other two have quite messy eigenvectors (I have computed them but I can't find any nice way to simplify them in a way akin to (3)).
(C) What meaning can we obtain from the eigenvectors and eigenvalues of the adjacency matrix? Is there some more intuitive connection between the flat eigenvalue and the localized states discussed in the paper?
All recommendations for places where I can read more about this and similar topics would be appreciated as well.