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Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics

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Materials science and the study of the electronic properties of solids are a major field of interest in both physics and engineering. The starting point for all such calculations is single-electron, or non-interacting, band structure calculations, and in the limit of strong on-site confinement this can be reduced to graph-like tight-binding models. In this context, both mathematicians and physicists have developed largely independent methods for solving these models. In this paper we will combine and present results from both fields. In particular, we will discuss a class of lattices which can be realized as line graphs of other lattices, both in Euclidean and hyperbolic space. These lattices display highly unusual features including flat bands and localized eigenstates of compact support. We will use the methods of both fields to show how these properties arise and systems for classifying the phenomenology of these lattices, as well as criteria for maximizing the gaps. Furthermore, we will present a particular hardware implementation using superconducting coplanar waveguide resonators that can realize a wide variety of these lattices in both non-interacting and interacting form.

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Acknowledgements

We thank David Huse, János Kollár, Charles Fefferman, Siddharth Parameswaran, and Péter Csikvári for helpful discussions.

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Correspondence to Alicia J. Kollár.

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Communicated by H. T. Yau.

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This work was supported by the NSF, the Princeton Center for Complex Materials DMR-1420541, and by the MURI W911NF-15-1-0397.

Appendices

Construction of \(\,\bar{\varvec{\!H}}_{\varvec{a}} (\varvec{X})\) from \(\varvec{N}\) and \(\varvec{N^t}\)

The incidence matrices M, \(M^t\), N, and \(N^t\) can be understood as instructions for how to navigate a graph X. Consider first the simpler case of half-wave modes, and begin with a state \(\mathbf {j}\) on the graph X which is one on the \(j^{th}\) vertex and zero elsewhere. The action of M on this state takes it to all the edges which are incident on \(v_j\). Acting next with \(M^t\) then goes from these incident edges back to the vertex set and lands on all vertices that touch these chosen edges, including the original source vertex \(v_j\). Therefore, \(M^t M\) describes how to get from a vertex \(v_j\) to neighboring vertices by taking a walk though the edge set; however, it includes the option of going back to the original site. Such an option is not allowed under the action of \(A_X\), but the number of times that this mistaken path can occur is precisely the degree of the vertex \(v_j\). It therefore follows that \(A_X = M^t M - D_X \).

Instructions for navigating the edge set can derived in an analogous manner. Starting from a state \(\mathbf {e}_j\) on \({\mathcal {E}}(X)\) which is one on the \(j^{th}\) edge and zero elsewhere, the action of \(M^t\) takes this state to all the vertices which are incident on the jth edge. Acting next with M then goes from these vertices to all the edges that they touch. Analogously to above, \(MM^t\) describes how to navigate from an edge to neighboring edges by taking a walk though the vertices, but once again, it allows for the option of returning to the original location. Since each edge has two ends, this incorrect path will occur exactly twice, and we find that \(A_{L(X)} = {\bar{H}}_s(X) = MM^t - 2 I\).

In the oriented half-wave case, we must navigate with N and \(N^t\) instead. The combinatorics of the allowed paths is identical to the full-wave case with M and \(M^t\). The only outstanding detail is the additional minus signs. Consider \(N^tN\). It allows two types of paths through the vertex set: correct paths to neighboring vertices and extra paths back to the source vertex. A path from the source vertex back to itself will always consist of hopping to one end of an edge and back off of that same edge. It will therefore carry an amplitude \((\pm 1)^2 = 1\). Moving from a vertex to its correct neighbors will always involve entering at one end of an edge and exiting at the other, and will therefore have the amplitude \(1 \times -1 = -1\). It therefore follows that \(A_X = D_X - N^tN\).

Now consider navigating the edge set using \(N N^t\). There are once again incorrect paths that return to the source edge, which will always have positive amplitude. Correct paths to neighboring vertices will have variable sign depending on which two types of edge ends are involved. If the transition goes between the ends which have the same sign, the transitions amplitude will be positive. Otherwise it will be negative. This is precisely the desired behavior for \({\bar{H}}_a(X)\), and it therefore follows that \({\bar{H}}_a(X) = NN^t - 2 I\).

Compact Support Eigenstates

\({\bar{H}}_s(X)\) has a compact support eigenstate with eigenvalue \(-2\) for every even cycle in X and \({\bar{H}}_a(X)\) has such a state for every cycle in X. To see this, we will demonstrate the compact support eigenstates by construction. Consider first the simpler case of \({\bar{H}}_s\), and assume that the layout graph X contains an even cycle. This will give rise to a cycle in L(X) with equal length. Choose a labeling of the vertices of X such that the even cycle is the first 2n vertices. Letting \(\oplus \) and \(\ominus \) denote addition and subtraction modulo 2n, the corresponding cycle in L(X) is then indexed by the unordered pairs of sequential vertices \(\{x,y\} = \{y,x\}\) such that \(y\equiv x \oplus 1 \). Define a state on L(X)

$$\begin{aligned} \psi _c(\{x,y\} ) = {\left\{ \begin{array}{ll} 1, &{} \text{ if } x,y \le 2n \text{, } x \text{ is } \text{ even } \text{ and } y \equiv x \oplus 1, \\ -1 &{} \text{ if } x,y \le 2n \text{, } x \text{ is } \text{ odd } \text{ and } y \equiv x \oplus 1 ,\\ 0 &{} \text{ otherwise }. \end{array}\right. } \end{aligned}$$
(91)

The state \(\psi _c\) obeys

$$\begin{aligned}&{\bar{H}}_s(X) \psi _c(\{ x,y\} )\\&\quad = {\left\{ \begin{array}{ll} \psi _c( \{ x\ominus 1,x\} ) + \psi _c(\{ x\oplus 1,x \oplus 2 \} ), &{} \text{ if } x,y \le 2n \text{ and } y \equiv x\oplus 1, \\ \psi _c( \{ x \ominus 1,x \}) + \psi _c(\{ x, x \oplus 1 \} ) &{} \text{ if } x \le 2n \text{ and } y> 2n.\\ 0 &{} \text{ if } x,y > 2n. \end{array}\right. } \end{aligned}$$

Using the oscillations in \(\psi _c(x,y)\), we find \({\bar{H}}_s \psi _c(x,y) = -2 \psi _c(x,y)\). The state \(\psi _c\) is therefore an eigenstate with eigenvalue \(-2\) which is localized and of compact support by construction.

Such compact-support eigenstates are highly unusual in generic lattices, but in line-graph lattices they arise due to destructive interference in the plaquettes which surround the vertices of the layout graph because hopping from neighboring sites of opposite sign cancels. In circuit QED lattices this same phenomenon can also be understood at the hardware level as destructive interference between voltages incident on the coupling capacitors at the vertices of the layout. If two of the ports of the coupler have equal and opposite voltage, then the outgoing voltage at the third port is their sum, which is zero. Examples of these compact support eigenstates are shown in Fig. 16a, c. In the absence of even cycles, the states with energy \(-2\) are believed to be exponentially localized, as was shown for the Cayley graph of \({\mathbb {Z}}_2 *{\mathbb {Z}}_3\) in Ref. [32].

The argument for \({\bar{H}}_a\) is similar, but is simplest in a well chosen orientation of X. Assume now that X contains a cycle of any length, even or odd. As before, this will give rise to a cycle in L(X) of equal length. Consider then states on this cycle. The analysis in this case is complicated by the need to chose an orientation of X and work in that particular realization of \({\bar{H}}_a(X)\), but the underlying physics of destructive interference is the same. To see this, choose a gauge where \(\varphi \) goes from negative to positive when moving around the cycle in a clockwise direction. This makes \(t_{i,j}\) everywhere positive going around cycle. Resonators that touch the cycle will see one site on the cycle with \(\varphi = 1\) and one with \(\varphi = -1\), so the \(t_{i,j}\) for leaving cycle will come in pairs, one positive and one negative. Consider the state \(\psi = 1,1,1,1, \cdots \) around the cycle and zero everywhere else. The tunneling amplitudes onto all neighboring resonators cancel by destructive interference, so \(\psi \) is perfectly localized. It therefore follows that

$$\begin{aligned} {\bar{H}}_a \psi (i) =2 \psi (i) = {\left\{ \begin{array}{ll} 2 \psi (i), &{} \text{ for } i \text{ in } \text{ the } \text{ cycle } , \\ 0 &{} \text{ for } i \text{ not } \text{ in } \text{ the } \text{ cycle } . \end{array}\right. } \end{aligned}$$

To prove the existence of a flat band, we need to show that construction of such a localized state can be done simultaneously for many (if not all) cycles in X. It is not generally possible to chose a gauge which is as simple as the example above for all cycles simultaneously, but it is not necessary. For any given cycle, the gauge choice described above demonstrates the existence of a localized tight-binding wavefunction with eigenvalue \(-2\). Since the tight-binding model is shorthand for the underlying voltage model in Eq. 13, the voltage configuration to which this state corresponds exists regardless of the choice of gauge. Therefore, a localized state of compact support on \({\mathcal {E}}(X)\) which is an eigenstate of \({\bar{H}}_a(X)\) exists for every cycle in X, and all 3-regular half-wave lattices have a flat band whether they are Euclidean or hyperbolic; or bipartite, or not. Examples of these states for two non-bipartite examples are shown in Fig. 16b, d alongside their full-wave counterparts.

Reduced \(C^*\)-algebras

In lieu of the band structures coming from Bloch waves in the Abelian setting (see Sect. 5) the theory of reduced \(C^*\)-algebras yields some information on the bands and gaps. For G a discrete (finitely-generated) torsion-free group (that is it has no nontrivial elements of finite order), the Kaplansky-Kadison conjecture asserts that the \(\ell ^2\) reduced group \(C^*\)-algebra \(C_{red}^* (G)\) has no nontrivial idempotents. More precisely, viewing elements \(\sum _{g\in G}{c_g g}\) (\(c_g \in {\mathbb {C}}\) and all but finitely many \(c_g\)’s being zero) in the convolution group algebra \(C^* (G)\) as bounded operators on \(\ell ^2 (G)\) by convolving on the left, \(C_{red}^* (G)\) is the closure of these operators in the operator norm of \(\ell ^2 (G)\). The Kaplansky-Kadison conjecture is known for many such G’s and in particular all of the G’s encountered in the paper (see Ref. [38]). An immediate consequence is that the spectrum of any \({\mathcal {D}}\) in \(C^* (G)\) acting on \(\ell ^2 (G)\) is connected and in particular, if \({\mathcal {D}}\) is self-adjoint then its spectrum is a single (possibly degenerate) interval. The proof of this idempotent property proceeds by showing that for any idempotent e, the (normalized) trace of e is integral and, hence, if it is not 0, then it must be 1, and \(e = I\). The integrality of the trace extends to \(C_{red}^* (G)\otimes \text{ Mat }_n({\mathbb {C}})\), where the trace on the second factor is the usual one on \(n\times n\) matrices with complex coefficients. It follows that the spectrum of any self-adjoint \({\mathcal {D}}\) in \(C_{red}^* (G)\otimes \text{ Mat }_n({\mathbb {C}})\) consists of at most n bands.

Fig. 16
figure 16

Flat-band states in full-wave and half-wave lattices. a The smallest flat band state in the heptagon-kagome lattice obtained as the line graph of hyperbolic heptagon-graphene (heptagonal honeycomb) lattice. All hopping matrix elements are negative and shown in light blue. The state is completely alternating and exists on an even cycle enclosing two plaquettes. b The smallest flat band state in the heptagon-kagome lattice using half-wave modes. Since the layout is non-bipartite, this Hamiltonian has both positive (dark blue) and negative (light blue) hopping matrix elements. This tight-binding wavefunction encloses a single odd cycle, and the orientation has been chosen to make it particularly simple. c A smallest flat band state in a Euclidean tiling obtained as the line graph of 3-regular tiling of heptagons and pentagons, known as the heptagon-pentgon-kagome lattice. As in its hyperbolic counterpart, this flat band state is alternating and encloses two plaquettes. d A smallest flat band state in the heptagon-pentagon-kagome lattice using half-wave modes. This choice of gauge necessitates a sign flip of the tight-binding wavefunction from one side of the plaquette to the other, but this state still encloses only a single plaquette

Following Sunada [43] we apply this to our infinite layouts X on which a torsion free G acts as automorphisms with \(|G \backslash V(X)| = n\). In this case we can realize \(A_X\) with its action on \(\ell ^2(X)\) as an element of \(C_{red}^* (G)\otimes \text{ Mat }_n({\mathbb {C}})\) and conclude that \(\sigma (A_X)\) has at most n bands. For example, if \(G = F_k\) the free group on generators \(g_1,g_2, \ldots g_k\); \({\mathcal {D}}= g_1 + g_1^{-1} + \cdots g_k + g_k^{-1}\); and X is the 2k-regular tree realized as the Cayley graph of G with generators \(g_1, g_1^{-1}, \ldots g_k, g_k^{-1}\); then \({\mathcal {D}}\) is identified with \(A_X\). According to the above \(\sigma (A_X)\) consists of one band. As we have noted before, according to Kesten [22] \(\sigma (A_X) = [-2\sqrt{2k-1}, 2\sqrt{2k-1}]\). Another example is \(X = {\mathbb {M}}\), the McLaughlin graph from Ref. [32] and Sect. 4. The group \(G = {\langle } R{\rangle } * {\langle } Q {\rangle }\) with \(Q^2 = R^3 = 1\) acts simply transitively on \(V({\mathbb {M}})\). The kernel \(\varGamma \) of \(\phi : G \rightarrow {\mathbb {Z}}/2{\mathbb {Z}} \times {\mathbb {Z}}/3{\mathbb {Z}}\) where \(Q\rightarrow (1,0)\), \(R\rightarrow (0,1)\) has index 6 in G and is torsion free. Hence \(|\varGamma \backslash {\mathbb {M}} |= 6\), and we can apply the idempotent theorem to conclude that \(A_{\mathbb {M}}\) has at most 6 bands. In fact, as McLaughlin showed, \(A_{\mathbb {M}}\) has 4 bands. Without a further understanding of the projections this technique of embedding \({\mathcal {D}}\) in such \(C^*\)-algebras gives upper bounds on the number of bands; bounds which we do not expect to be sharp.

Fig. 17
figure 17

Fundamental domains and symmetry generators. Schematic drawing of a single plaquette of \(T_k\), showing the two hyperbolic triangles \(\blacktriangle _k\) and \(\blacktriangle _k^\prime \) whose reflections generate the symmetry group \(G_k\), and whose union constitutes the fundamental domain for the largest torsion-free subgroup \(\varGamma _k\) in \(G_k\)

We apply this to the tessellation graphs \(T_k\), \(k \ge 7\) for which this \(C^*\)-algebra method is the only one that we know of that controls the number of bands. The full symmetry group \(G_k\) of \(T_k\) is the Coxeter reflection group [k, 3] that is the reflection group of a hyperbolic triangle \(\blacktriangle _k\) with angles \((\pi /2, \pi /3, \pi /k)\), shown in Fig. 17. \(G_k\) has a presentation with generators \(R_1,R_2,R_3\) and relations \(R_1^2 = R_2^2 = R_3^2 = (R_1 R_2)^2 = (R_2 R_3)^3 = (R_3 R_1)^k = 1\). The index-2 subgroup \({\mathcal {D}}_k\) of \(G_k\) consisting of even words in the R’s, and consists of the orientation preserving isometries of the hyperbolic plane \({\mathbb {H}}\) that preserve \(T_k\). \({\mathcal {D}}_k\) acts transitively on \(V(T_k)\) with the stabilizer of any \(v \in V(T_k)\) being order 3. \({\mathcal {D}}_k\) acts on \({\mathbb {H}}\) with a fundamental domain \(\blacktriangle _k \cup \blacktriangle ^\prime _k\) of area \((k-6) \pi /3k\) (shown in Fig. 17) and it has a presentation with generators ABC and relations

$$\begin{aligned} A^2 = B^3 = C^k = ABC = 1. \end{aligned}$$
(92)

In order to apply the \(C^*\)-algebra results above we seek the largest torsion-free subgroup \(\varGamma _k\) of \({\mathcal {D}}_k\). If \(\varGamma \) is such a group of index \(m = m_\varGamma \), then the compact orientable hyperbolic surface \(\varGamma \backslash {\mathbb {H}}\) has genus \(g = g_\varGamma \ge 2\). The Gauss-Bonnet formula relates the genus to its area.

$$\begin{aligned} \text{ Area }(\varGamma \backslash {\mathbb {H}} )= 4 \pi (g_\varGamma - 1). \end{aligned}$$
(93)

On the other hand, a fundamental domain for the action of \(\varGamma \) on \({\mathbb {H}}\) consists of m copies of \(\blacktriangle _k \cup \blacktriangle _k ^\prime \). The area of \(\blacktriangle _k\) is \((\pi - \pi /2-\pi /3 - \pi /k)\) which leads to the relation

$$\begin{aligned} m(k-6) = 12(g-1). \end{aligned}$$
(94)

The smallest integer solution to Eq. 94 depends on the factorization of k and is given by \(m_k\) in Table 2 assuming the factorization \(k = 2^a 3^b k_1\), with \(k > 6\) and \(k_1 \equiv 1 \mod 6\).

Table 2 Genus \(g_k\) and index \(m_k\) of the largest torsion-free subgroup \(\varGamma _k\) in \({\mathcal {D}}_k\) versus the exponents a and b of the factorization of k

Thus, the smallest index of a \(\varGamma \) in \({\mathcal {D}}_k\) which is torsion free is at least \(m_k\). In fact, one can find a torsion free \(\varGamma _k\) in \({\mathcal {D}}_k\) with index \(m_k\) (see Ref. [13] where this smallest index is resolved for a Fuchsian group).

Now \(\varGamma _k\) acts freely on \(V(T_k)\) and since the stabilizer of any \(v \in V(T_k)\) has order 3 in \({\mathcal {D}}_k\), it follows that the number of orbits of \(\varGamma _k\) on \(V(T_k)\) is

$$\begin{aligned} |\varGamma _k \backslash V(T_k)| = \frac{m_k}{3}. \end{aligned}$$
(95)

According to the idempotent results applied to \(C_{red}^* (\varGamma _k) \otimes \text{ Mat }_{m_k/3}({\mathbb {C}})\), we deduce that \(\sigma (T_k)\) has at most \(m_k/3\) bands.

The Gap at \(-2\) for \(\,\bar{\varvec{\!H}}_{\varvec{s}}(\varvec{X})\) and finite \(\varvec{X}\)

Our aim is to prove the lower bound of Eq. 34 which according to Eq. 24 amounts to giving a lower bound for \(\lambda _{\text {min}} \left( D_X + A_X \right) \) for a layout X. One could proceed as in the proof of Eq. 89, however there is an illuminating combinatorial characterization of \(\lambda _{\text {min}}\) being bounded from below (uniformly as \(|X| \rightarrow \infty \)) that we use instead. For \(S \subset V(X)\), let cut(S) be the set of edges of X with one end point in S and the other outside S. Let \(e_{\text {min}}(S)\) be the minimum number of edges of S that need to be removed (here S is the induced subgraph) so that the resulting graph is bipartite. Set

$$\begin{aligned} \digamma (X) = \min _{\phi \ne S \subset V(X)} \frac{e_{\min }(S) + |\text {cut} (S)|}{|S|}. \end{aligned}$$
(96)

The following is a bipartite analogue of the combinatorial Cheeger inequality [2] and is formulated and proved in Ref. [11]

$$\begin{aligned} \frac{\digamma ^2(X)}{4 \, d^{*}(X)} \le \lambda _{\text {min}} (D_X + A_X) \le 4 \digamma (X). \end{aligned}$$
(97)

where \(d^{*}(X)\) is the maximum degree of any vertex of X.

We use this to prove that if \(r \ge 2\) is fixed and X is any layout for which the induced subgraphs on \(B_r (x) = \left\{ y \in X : d(y, x) \le r \right\} \) are non-bipartite for all \(x \in X\), then

$$\begin{aligned} \lambda _{\text {min}} (D_X + A_X) \ge \left[ 48 \left( 3.2^{2 r -1} - 1 \right) ^2 \right] ^{-1}. \end{aligned}$$
(98)

To apply Eq. 97 we estimate \(\digamma (X)\) for X’s which satisfy this local non-bipartite condition. Let \(S \subset V(X)\) and choose \(W \subset S\) a maximally 2r-separated subset, that is for \(v, w\in W\)\(d_X (v, w) > 2r\). Then

$$\begin{aligned} |W| \ge \frac{|S|}{M_{2r}}, \ \ \text {with} \ \ M_t = 3.2^{t} - 2. \end{aligned}$$
(99)

If this is not true then

$$\begin{aligned} |W|< \frac{|S|}{M_{2r}}, \ \ \text {with} \ \ |\bigcup \limits _{w\in W} B_{2r}(w)|< |W| M_{2r} < |S|, \end{aligned}$$
(100)

where we have used the fact that for a layout \(|B_t (x)| \le M_t\). In this case there would exist \(s \in S\) such that \(s\notin B_{2r} (w)\) for any \(w \in W\), but then \(s \cup W\) is 2r-separated and larger than W, which contradicts the assumption that W was maximal. Thus, Eq. 99 holds.

For \(w \in W\) either

  1. (i)

    \(B_r (w) \cap S = B_r (w)\), in which case this induced subgraph is non-bipartite by assumption, and hence this local contribution to \(e_{\text {min}}(S)\) in Eq. 96 is at least 1, or

  2. (ii)

    \(B_r(w)\cap S \subsetneqq B_r(w)\) in which case the local contribution to \(\text {cut}(S)\) is at least 1, and this edge is in \(B_r(w)\).

Thus, in either case the local contribution to the numerator in Eq. 96 is at least 1. Since the different \(B_r(w)\)’s with \(w\in W\) are disjoint it follows from above that

$$\begin{aligned} \digamma (X) \ge \frac{|S|}{M_{2r}|S|} = (M_{2r})^{-1} \end{aligned}$$
(101)

which establishes Eq. 98.

Classification of Hoffman Layouts

X is a layout (i.e. has degree at most 3) and \(A_X\) its adjacency operator. Thanks to the results of Refs. [8, 12, 19], it is known that if \(\lambda _{\text {min}}(A_X)> -2\), then X is either a generalized line graph (see below for the definition) \(L(T; 1, 0, \ldots , 0)\) with T a tree; a line graph \(L({\mathcal {K}})\) with \({\mathcal {K}}\) a tree or an odd cycle; or it is one of a finite list of X’s (see Theorem 2.1 of Ref. [12]). In the case that \(X = L({\mathcal {K}})\) and \({\mathcal {K}}\) is a tree, it follows from Eq. 24 that \(\sigma (A_X) = -2 + \sigma (D_{\mathcal {K}} + A_{\mathcal {K}})\) and since \({\mathcal {K}}\) is bipartite \(\sigma (A_X) = -2 + \sigma (D_{\mathcal {K}} - A_{\mathcal {K}})\). Moreover, \({\mathcal {K}}\) is planar and of bounded degree and hence from Ref. [29] it follows that

$$\begin{aligned} \lim _{|{\mathcal {K}}|\rightarrow \infty } \lambda _{\text {min}}\left( A_{L({\mathcal {K}})} \right) \le -2. \end{aligned}$$
(102)

In as much as T is an induced subgraph of \(L(T; 1, 0,\ldots ,0)\) it follows that Eq. 102 continues to hold for \(X = L(T; 1, 0,\ldots ,0)\) when \(|T|\rightarrow \infty \). Equation 102 also holds for \({\mathcal {K}}\) an odd cycle and \(|{\mathcal {K}}|\rightarrow \infty \). Putting these together we deduce that for X a layout

$$\begin{aligned} \limsup _{|X| \rightarrow \infty } \lambda _{\text {min}} (A_X) \le -2. \end{aligned}$$
(103)

Write \({\mathbb {E}}_X = 3 I - D_X\), then \({\mathbb {E}}_X\) is diagonal with nonnegative entries and hence

$$\begin{aligned} \lambda _{\text {min}}(D_X + A_X)= & {} \lambda _{\text {min}} (3I - {\mathbb {E}}_X + A_X) \end{aligned}$$
(104)
$$\begin{aligned}\le & {} \lambda _{\text {min}} (3I + A_X) \le 3 + \lambda _{\text {min}}(A_X). \end{aligned}$$
(105)

Therefore from Eq. 103

$$\begin{aligned} \limsup _{|X|\rightarrow \infty } \lambda _{\text {min}} (D_X + A_X) \le 3-2 = 1. \end{aligned}$$
(106)

From Eq. 24 this gives that

$$\begin{aligned} \limsup _{|X|\rightarrow \infty } \lambda ({\bar{H}}_s (X)) \le -1, \end{aligned}$$
(107)

proving Eq. 35.

For the rest of this Appendix, X is a 3-regular layout. For these one can strengthen Eq. 103 to (see Thm. 2.5 in Ref. [12]):

$$\begin{aligned} \lambda _{\text {min}} (A_X) \le -2, \end{aligned}$$
(108)

except for the case that X is the complete graph \(T_3\). The case that \(\lambda _{\text {min}}(A_X) = -2\), that is when X is a 3-regular Hoffman graph, can be characterized. According to Ref. [8], any such graph is one of a finite number of graphs or a generalized line graph. In more detail the latter are given as follows: there is a connected graph Y with n vertices \(v_1,v_2,\ldots ,v_n\) and non-negative integers \(a_1,a_2,\ldots ,a_n\) and the “cocktail party” graphs \(CP(a_j)\) with \(2a_j\) vertices and degree \(2a_j -2\). (CP(0) is the empty set, , , ...) The generalized line graph \(X = L(Y; a_1,\ldots ,a_n)\) is defined by: X has as its vertices those of L(Y) as well as those of \(CP(a_1)\),..., \(CP(a_n)\), and also all the edges of these graphs together with extra edges joining any edge e of Y (i.e. a vertex of L(Y)) to all the vertices of \(CP(a_j)\) if \(v_j\) is an end point of e.

In order that \(L(Y; a_1, \ldots , a_n)\) be 3-regular we must have \(a_j = 0\) or 1 for all j. For if \(a_j \ge 2\) for some j then any edge emanating from \(v_j\) will have degree bigger than 3 in X. Actually, \(a_j = 1\) is also impossible since and if \(v_j \in Y\) has degree 3 or more than any edge e emanating from \(v_j\) would have degree at least 4 in X. On the other hand, if \(v_j\) in Y has degree 1 or 2 then the vertices of X corresponding to \(CP(a_j)\) have degree less than three. It follow that all the \(a_j\)’s are 0 and hence a large 3-regular Hoffman graph X is of the form \(X = L(Y)\). In order that X be 3-regular one checks that Y has to be 3, 2-biregular. A 3, 2-biregular graph Y is obtained as a subdivision graph of a 3-regular graph Z. We conclude that a 3-regular Hoffman graph X is of the form

$$\begin{aligned} X = L({\mathbb {S}}(Z)) \text{ for } \text{ some } \text{3-regular } Z. \end{aligned}$$
(109)

This competes the proof of the claims in Sect. 4.1 that all regular Hoffman layouts are achieved by the process leading to Eq. 36.

We conclude this Appendix with proofs of Eqs. 38 and 39. Firstly, if X is \([-3,-2)\)-gapped and is large, then according to the discussion above, \(X = L({\mathbb {S}}(Z))\) for a cubic Z. From the Eq. 115 in “Appendix F” it follows that the rest of the gap set in Eq. 38 corresponds exactly to Z being \([-3,-2\sqrt{2}) \cup (2 \sqrt{2}, 3]\)-gapped. Moreover, non-bipartite Ramanujan Z’s achieve these gaps. Equation 38 then follows by combining these observations. Equation 39 is obtained similarly by choosing Z to be a Hoffman graph.

Subdivision Graphs and Their Line Graphs

Let X be a regular graph of degree three with \(n = 2\nu \) vertices. Its subdivision graph \({\mathbb {S}}(X)\) is a 3, 2-biregular graph with \(m = 3\nu \) vertices of degree 2 and \(n = 2\nu \) vertices of degree 3 with \(6\nu \) edges. Next, form the line graph \(L({\mathbb {S}}(X))\). It is a cubic graph with \(6\nu \) vertices. The adjacency matrices of these three graphs are closely related and their characteristic polynomials can be related to one another using block matrix identities. The treatment we give here is a summary of the proofs given Chapter 2 of Ref. [10], specializing to the case of a starting graph of degree three.

Let M and N be the incidence operators of the starting graph X as defined in Sect. 4. The adjacency matrices of X, \({\mathbb {S}}(X)\) and \(L({\mathbb {S}}(x))\) can be written as

$$\begin{aligned} A_X= & {} M^t M - D_X, \nonumber \\ A_{{\mathbb {S}}(X)}= & {} \begin{bmatrix} 0&M \\ M^t&0, \end{bmatrix} \end{aligned}$$
(110)
$$\begin{aligned} A_{L({\mathbb {S}}(X))}= & {} \begin{bmatrix} M \left( \frac{M^t + N^t}{2} \right) - I_{3\nu }&I_{3\nu } \\ I_{3\nu }&M \left( \frac{M^t - N^t}{2}\right) - I_{3\nu }, \end{bmatrix} \end{aligned}$$
(111)

where \(I_{l}\) is the \(l\times l\) identity matrix.

Two lemmas from linear algebra are required to relate the characteristic polynomials of these matrices. First, given an \(m \times n\) matrix B, the characteristic polynomials of \(B^t B\) and \(B^t\) are related by

$$\begin{aligned} \lambda ^n P_{BB^t}(\lambda ) = \lambda ^m P_{B^t B}(\lambda ). \end{aligned}$$
(112)

Second, given a \(2\times 2\) block matrix with square diagonal blocks, its determinant can be computed from the blocks if one of the diagonal blocks is non-singular:

$$\begin{aligned} \left| \begin{bmatrix} B_1&B_2 \\ B_3&B_4, \end{bmatrix} \right| = |B_1| \times |B_4 - B_3 B_1^{-1}B_2|. \end{aligned}$$
(113)

Applying these identities to X, \({\mathbb {S}}(X)\), and \(L({\mathbb {S}}(X))\) yields three relations.

$$\begin{aligned} P_{{\mathbb {S}}(X)}(\lambda ) = (\lambda )^\nu \times P_{X}(\lambda ^2 -3), \end{aligned}$$
(114)

and

$$\begin{aligned} P_{L({\mathbb {S}}(X))}(\lambda )= & {} (\lambda +2)^\nu \times (\lambda )^\nu \times P_{X}(\lambda ^2 -\lambda - 3)\nonumber \\= & {} P_{L(X)}(\lambda ^2 -\lambda - 2). \end{aligned}$$
(115)

These in turn can be converted to the eigenvalue relations

$$\begin{aligned} E_{{\mathbb {S}}(X)}= {\left\{ \begin{array}{ll} \pm \sqrt{E_X +3}\\ 0, \end{array}\right. } \end{aligned}$$
(116)

and

$$\begin{aligned} E_{L({\mathbb {S}}(X))}= & {} {\left\{ \begin{array}{ll} \frac{1 \pm \sqrt{1 + 4(E_X + 3)}}{2}\\ 0,\\ -2 \end{array}\right. } \nonumber \\= & {} \frac{1 \pm \sqrt{1 + 4(E_{L(X)} + 2)}}{2}. \end{aligned}$$
(117)

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Kollár, A.J., Fitzpatrick, M., Sarnak, P. et al. Line-Graph Lattices: Euclidean and Non-Euclidean Flat Bands, and Implementations in Circuit Quantum Electrodynamics. Commun. Math. Phys. 376, 1909–1956 (2020). https://doi.org/10.1007/s00220-019-03645-8

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