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

Alicia J. Kollár, Mattias Fitzpatrick, Peter Sarnak, Andrew A. Houck

arXiv:1902.02794v2quant-phcond-mat.othermath-phmath.COmath.MP

TL;DR

The paper addresses how to understand the unusual spectra of line-graph tight-binding lattices by combining physics and graph theory. It derives spectral and gap results for these lattices and examines their realization with circuit-QED resonators.

  • Problem

    Physics and mathematics offer complementary methods for tight-binding models, and the paper investigates how they can characterize line-graph lattices with unusual spectra.

    Source: §VII · p. 31

  • Method

    The paper relates circuit-QED resonator lattices to line-graph operators and applies graph-theoretic and band-structure analyses across Euclidean and non-Euclidean layouts.

    Source: §VII · pp. 31–32

  • Results

    The effective operators have flat bands at −2 for finite or infinite, regular or irregular, homogeneous or inhomogeneous layouts, and the paper derives criteria for spectral gaps.

    Source: §VII · p. 31

  • Takeaways & Limitations

    Isolated flat bands with added photon interactions are a proposed starting point for quantum simulation of strongly correlated many-body physics with photons.

    Source: §VII · p. 32

Abstract

from arXiv · show

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.

I. INTRODUCTION

The paper combines solid-state physics and graph theory to analyze line-graph lattices with unusual band structures. It introduces tight-binding models and graph concepts used in that analysis.

I. INTRODUCTION

  • The paper combines physics methods with graph-theory results to analyze line-graph lattices and prove relevant corollaries about their unusual band structures.Its focus includes spectrally isolated flat bands and applications to Euclidean and non-Euclidean solids.Source: §I · pp. 2–3
  • In the tight-binding limit, well-separated bound states provide a discrete approximation to the continuum problem, with each site amplitude weighting its on-site wavefunction.The approximation is motivated by weak kinetic energy relative to the lattice potential.Source: §I · p. 2
  • Neighboring-site wavefunction overlap produces hopping, so the low-order Hamiltonian is dominated by nearest-neighbor couplings.The bound-state energy shift is a constant offset that can be set to zero in the single-band approximation.Source: §I · p. 3
  • Flat-band lattices are studied for their sensitivity to interactions, and superconducting circuits offer a hardware setting for examining interacting photon models.The paper derives gap criteria and maximum-size bounds, identifies examples attaining them, and examines finite hard-wall boundaries.Source: §I · p. 3

II. THE TIGHT-BINDING HAMILTONIAN AND THE GRAPH LAPLACIAN

  • A lattice associates points with hopping elements, while its graph represents sites as vertices and nonzero hoppings as edges.For uniform hopping, the chosen nonzero elements define nearest neighbors; graph adjacency and the graph Laplacian provide related descriptions of the Hamiltonian.Source: §II · p. 4

III. CIRCUIT QED LATTICES

Coupled superconducting CPW resonators provide a circuit-QED implementation of tight-binding lattices. Their resonant modes yield effective hopping models whose signs depend on mode orientation.

A. Tight-Binding Solids in Circuit QED

  • The resonators support symmetric full-wave and antisymmetric half-wave modes, with standing-wave functions and antinodes at their ends.The two modes have β = 1 and β = −1, respectively.Source: A. Tight-Binding Solids in Circuit QED · p. 5
  • Coupled CPW resonators act as artificial tight-binding materials, with resonator modes replacing on-site states and microwave photons replacing electrons.The coupling terms are derived from the transmission-line Lagrangian.Source: A. Tight-Binding Solids in Circuit QED · pp. 4–5
  • The effective Hamiltonian is a tight-binding model with one site per resonator and hopping only between resonators that share endpoints.Eliminating generalized fluxes in favor of creation and annihilation operators gives a computationally more convenient form.Source: A. Tight-Binding Solids in Circuit QED · p. 5
  • The device has a layout graph with capacitors as vertices and resonators as edges, while the effective graph uses resonators as vertices connected when they touch.Effective-lattice wavefunctions specify flux at a chosen resonator end and encode flux along the resonator.Source: A. Tight-Binding Solids in Circuit QED · p. 6
  • For half-wave modes, reversing relative on-site wavefunction signs changes the hopping sign, and non-bipartite graphs prevent choosing all couplings with the same orientation.Different orientation choices can describe the same generalized flux while changing the tight-binding wavefunction representation.Source: A. Tight-Binding Solids in Circuit QED · p. 6, Figure 1 · p. 6

B. Full-Wave Versus Half-Wave Models: Line-Graph Effective Lattices

The section examines how full-wave and half-wave modes produce line-graph effective lattices, including uniform- and signed-hopping models, and relates layout-graph properties to spectral gaps. It also identifies graph examples associated with a gap interval and a locally maximal planar gap interval.

B. Full-Wave Versus Half-Wave Models: Line-Graph Effective Lattices

  • Full-wave modes yield a line-graph lattice with equal, negative hopping and an infinitely degenerate flat band at −2.The effective lattice sites are the edges of the layout graph, and the hopping magnitudes do not depend on nearest-neighbor distances.Source: B. Full-Wave Versus Half-Wave Models:… · pp. 7–8
  • Half-wave and full-wave models coincide on bipartite layouts; on non-bipartite layouts, half-wave modes produce signed hopping whose matrix depends on orientation but whose eigenvalues do not.The half-wave model is an inverted p-wave model with equal magnitudes for all nonzero hopping elements.Source: B. Full-Wave Versus Half-Wave Models:… · p. 8
  • On an odd cycle, full-wave modes retain the eigenvalue +2 but lose −2, whereas half-wave modes retain −2 but lose +2.The missing alternating pattern is frustrated because the cycle is non-bipartite.Source: B. Full-Wave Versus Half-Wave Models:… · pp. 8–9, Figure 3 · p. 9

C. Flexible Resonators and Non-Euclidean Graphs

  • CPW resonators set site frequencies by resonator length and hopping by coupling-capacitor geometry, enabling layouts without equal vertex spacing; planar degree-3 graphs are emphasized for fabrication robustness.Non-planar layout graphs require multilayer fabrication, while higher coordination can produce asymmetric capacitors and unequal hopping.Source: C. Flexible Resonators and… · pp. 9–10

D. Sample Sizes

  • Finite hard-wall truncations can introduce states absent from the infinite-lattice spectra, and boundary effects are especially consequential for hyperbolic graphs.For hyperbolic truncations, the boundary is a finite fraction of the total volume.Source: D. Sample Sizes · p. 10

A. Finite Layouts

  • For 3-regular layouts, the spectra reduce to adjacency spectra, with −2 multiplicities m−n+1 for H_a and m−n+η for H_s.Here η is 1 for bipartite layouts and 0 otherwise; the two models have identical spectra when the layout is bipartite.Source: A. Finite Layouts · p. 11
  • Planar layouts cannot maintain a gap above −2 for H_a as their size grows, while H_s has a uniform gap when every fixed-radius neighborhood is non-bipartite.The local condition applies for a fixed radius r ≥ 2.Source: A. Finite Layouts · pp. 11–12
  • Hoffman layouts are defined by attaining the stated bound on the gap at −2 for H_s.The paper uses a graph classification to identify layouts that attain equality in that bound.Source: A. Finite Layouts · p. 12
  • The Cayley graph of Z2 ∗Z3 realizes the cited gap interval, and its line graph is shown in Fig. 6.The supplied passage does not specify the interval’s bounds clearly.Source: A. Finite Layouts · pp. 13–14
  • An example graph realizing a locally maximal planar gap interval is shown in Fig. 5.Source: A. Finite Layouts · p. 14

B. Infinite Regular Layouts

For infinite regular layouts, line-graph spectra are related directly to the layout graph’s adjacency spectrum, yielding criteria for flat-band gaps. Amenability, bipartiteness, and specific graph families determine when those gaps occur and how large they can be.

B. Infinite Regular Layouts

  • σ(Ḣ_s(X)) and σ(Ḣ_a(X)) consist of a flat band at −2 plus shifted or reflected copies of σ(A_X).The relation makes σ(A_X) the primary spectrum to analyze.Source: B. Infinite Regular Layouts · p. 14
  • The −2 eigenvalue has infinite multiplicity, while compactly supported eigenstates occur for Ḣ_s(X) exactly when X has an even non-backtracking circuit and for Ḣ_a(X) unless X is the 3-regular tree.The compactly supported states are constructed in Appendix A.Source: B. Infinite Regular Layouts · p. 14
  • For non-amenable G, −2 is gapped for both Ḣ_a(X) and Ḣ_s(X); for amenable G, it is ungapped for Ḣ_a(X) and gapped for Ḣ_s(X) exactly when X is non-bipartite.For non-amenable groups, the induced finite subgraphs also have −2 gapped in σ(Ḣ_s(X)).Source: B. Infinite Regular Layouts · p. 15
  • A 4-regular line graph can have a gap of 1 above its flat band, the maximum possible among 4-regular graphs.The cited graph and its line graph are universal covers for all 3-regular examples discussed in the paper.Source: B. Infinite Regular Layouts · p. 16
  • L(S(X)) produces flat bands at −2 and 0; for the 3-regular tree example, the −2 and 0 bands are gapped and the zero-band states are exponentially localized.The associated graph M is the Cayley graph of Z2 ⋆ Z3 and has two absolutely continuous spectral intervals in addition to its flat bands.Source: B. Infinite Regular Layouts · p. 16, Figure 6 · p. 16

A. Bloch Theory

Bloch theory parametrizes periodic-lattice eigenstates and energies by momentum, while incidence maps relate the band structures of a layout and its line graph. For Euclidean regular lattices, the line-graph bands include compactly supported flat-band states, with gap conditions depending on bipartiteness and the hopping model.

A. Bloch Theory

  • Periodic translations define the Bravais lattice, and Bloch’s theorem parametrizes the eigenstates and energy bands by momentum in the first Brillouin zone.When the unit cell contains multiple sites, the Bloch wave is vector-valued, with one entry per site.Source: A. Bloch Theory · pp. 17–18
  • For Euclidean d-regular lattices, the dispersive bands of Ḣ_s(X) and Ḣ_a(X) are determined by the bands of H_X, with additional flat bands at −2 consisting of compactly supported localized eigenstates.The relations follow by mapping layout eigenstates through the incidence matrices.Source: A. Bloch Theory · pp. 19–20
  • The incidence maps take a layout Bloch wave to line-graph Bloch waves with the same momentum, so the line-graph bands are shifted copies of the layout bands in the square-lattice and graphene examples.The line-graph bands shown include a flat band at −2.Source: Figure 8 · p. 18, A. Bloch Theory · p. 20
  • In Euclidean lattices, the flat bands of Ḣ_a(X) are never gapped, whereas those of Ḣ_s(X) are gapped exactly when X is non-bipartite.For Ḣ_a(X), the band is ungapped because H_X has states arbitrarily near d; for bipartite X, spectral symmetry likewise leaves Ḣ_s(X)'s flat band ungapped.Source: A. Bloch Theory · pp. 20–21

B. Real-Space Topology and Gapped Flat Bands at -2

On the kagome lattice, counting compact and noncontractible flat-band states forces a band touch. For other line graphs, overlap and cycle structure change that count and can permit gapped flat bands.

B. Real-Space Topology and Gapped Flat Bands at -2

  • The kagome lattice has M−1 independent plaquette states plus two noncontractible loops, forcing a dispersive band to touch the flat band at −2.The resulting M+1 states exceed the M states expected for one flat band.Source: Figure 9 · p. 21, B. Real-Space Topology and Gapped Flat… · p. 22
  • The topology argument generalizes to line graphs of 3-regular Euclidean lattices with only even cycles.The kagome lattice is the simplest example in this class.Source: B. Real-Space Topology and Gapped Flat… · p. 22
  • In a non-bipartite layout with odd smallest cycles, overlapping states encircling pairs of plaquettes alter the torus count and remove the requirement for a band touch.For the heptagon-pentagon kagome lattice, four independent such states per unit cell produce 4M flat-band states; the layout’s non-bipartite structure guarantees a gap.Source: B. Real-Space Topology and Gapped Flat… · p. 22
  • When the smallest localized states cover single plaquettes without overlap, the topology argument matches kagome and the flat bands at −2 cannot be gapped.This applies to ¯Hs(X) for bipartite X and to ¯Ha(X) for any X.Source: B. Real-Space Topology and Gapped Flat… · p. 22

C. Finite-Size Effects

Finite induced subgraphs introduce boundary effects into the spectrum. Periodic boundaries are generally incompatible with single-layer circuit-QED fabrication, while Euclidean geometry makes the edge-state fraction vanish as samples grow.

C. Finite-Size Effects

  • Finite experimental samples can be modeled by induced subgraphs S(X), formed by removing vertices outside a finite region, but their boundary coordination numbers vary.These graphs retain many properties and symmetries of the infinite lattice.Source: C. Finite-Size Effects · p. 22
  • Irregular boundaries create additional eigenvalues in ¯Hs(S(X)) and ¯Ha(S(X)) that do not belong to the infinite-lattice spectrum.Periodic boundaries can remove this issue theoretically but usually yield non-planar graphs incompatible with single-layer circuit-QED fabrication.Source: C. Finite-Size Effects · p. 22
  • For Euclidean lattices, edge states from hard-wall truncation make up a vanishing fraction of possible states as system size increases.Source: C. Finite-Size Effects · p. 22

D. Maximally Gapped Flat Bands

The section identifies layout-graph conditions for isolating and maximizing flat-band gaps, then illustrates them with constructed line-graph lattices. Hoffman-graph constructions attain maximal gaps in the examples described.

D. Maximally Gapped Flat Bands

  • For gapped flat bands at −2, the proposed Euclidean construction uses ¯Hs(X), a non-bipartite layout graph, and 3-regular layouts.The 3-regular restriction is used to keep the required all-way couplers physical.Source: D. Maximally Gapped Flat Bands · p. 23
  • The 4-regular line graph built from the constructed layout has maximally gapped flat bands at −2 and 1.Figure 12 presents its band structure and density of states.Source: D. Maximally Gapped Flat Bands · p. 24, Figure 12 · p. 25
  • A 3-regular Hoffman graph with least eigenvalue −2 yields a maximal gap of 1 above the flat band at −2.With finitely many exceptions, this least eigenvalue is achieved only by 3-regular line graphs.Source: D. Maximally Gapped Flat Bands · p. 24
  • Starting from a 3-regular graph X, the construction forms S(X) and then L(S(X)); using a Hoffman graph as the initial graph gives the largest gaps.The resulting Euclidean layout L(S(X)) is also 3-regular.Source: D. Maximally Gapped Flat Bands · pp. 24–25
  • For ¯Hs(X), the flat band at −2 is never gapped when X is bipartite, while the flat band at 0 is gapped exactly when X is non-bipartite.The gap at 0 depends on how non-bipartite X is; graphene gives no gaps, while heptagon-pentagon graphene gives large but non-maximal gaps.Source: D. Maximally Gapped Flat Bands · p. 25
  • The constructed graph X = L(S(X)) achieves maximal gaps at both −2 and 1 in ¯Hs(X).The supplied figure description reports a gap of 1 above −2 and identifies the band at 1 as ungapped in a separate graphene-based example.Source: D. Maximally Gapped Flat Bands · p. 25, Figure 12 · p. 25

VI. 3-REGULAR TESSELLATIONS OF REGULAR POLYGONS

The section examines spectra for trivalent graphs from regular-polygon tessellations, spanning finite spherical, Euclidean, and infinite hyperbolic cases. It reports finite-graph spectra and the infinite-dimensional flat-band eigenspace for T6.

VI. 3-REGULAR TESSELLATIONS OF REGULAR POLYGONS

  • The trivalent tessellations T3, T4, and T5 correspond to the tetrahedron, cube, and dodecahedron; T6 is the hexagonal Euclidean tessellation, and k≥7 gives infinite hyperbolic tessellations.Source: §VI · p. 25, Figure 13 · p. 26
  • The section analyzes ¯Hs(Tk) and ¯Ha(Tk), whose spectra lie within [−2, 4], to determine when they are gapped at −2.Source: §VI · pp. 25–26
  • For the finite examples T3 and T4, the paper lists spectra for the layout graphs and effective models, with multiplicities explained by dimensions of irreducible representations of Gk.Source: §VI · p. 26
  • For T6, the eigenvalue −2 has an infinite-dimensional space of ℓ2(E(T6)) eigenvectors, while the rest of the spectrum on (−2, 4) is absolutely continuous.The paper notes that Bloch waves can be used to compute the T6 spectrum.Source: §VI · p. 26

A. k ≥7: The Hyperbolic Cases

The hyperbolic layout spectra have finite-band structure and converge toward the 3-regular-tree spectrum as polygon size grows. The effective full-wave model has a gap at −2 for odd-sided layouts, unlike the half-wave model.

A. k ≥7: The Hyperbolic Cases

  • The spectrum of T_k consists of finitely many bands and converges to the spectrum of the 3-regular tree as k increases.The paper bounds the number of bands by m_k/3; for k = 7, 8, 9, 10, and 30, the bounds are 28, 16, 12, 10, and 10, respectively.Source: A. k ≥7: The Hyperbolic Cases · pp. 26–27
  • 2.862 ≤ λmax(T7) ≤ 2.966 and 2.852 ≤ λmax(T8) ≤ 2.943 are reported as tight bounds on the largest eigenvalues.These analytic bounds use the estimates in Equations 80 and 81.Source: A. k ≥7: The Hyperbolic Cases · p. 28
  • For odd k, the full-wave operator on induced balls is gapped at −2, whereas the half-wave operator’s gap vanishes as the radius grows.For even k, the full-wave gap also vanishes as the radius grows.Source: A. k ≥7: The Hyperbolic Cases · pp. 27–28
  • For the full-wave operator, −2 has multiplicity |E(S_r)| − |V(S_r)|, with a spectral gap above it for odd-sided layouts.The lower bound on the gap is derived by estimating a quotient; the passage does not provide its completed expression.Source: A. k ≥7: The Hyperbolic Cases · p. 28

B. Numerics

Numerical studies estimate the infinite-layout spectral gaps from truncated hyperbolic graphs and examine finite-size behavior. They find edge-induced states near −2 whose character depends on the model and layout.

B. Numerics

  • For r = 2, the upper adjacency gap increases monotonically with k toward ∼0.28, while the lower gap oscillates with bipartiteness.These small induced graphs remain far from the infinite-size limit; the lower gap is above its asymptotic limit when k is odd.Source: B. Numerics · p. 30
  • For k > 5, the lower gap of the full-wave model on S2(Tk) is smaller than the adjacency gap and decreases with k; the 1/k2 bound is not sharp.At k = 5, S2(T5) is the complete spherical tiling, so the two gaps coincide.Source: B. Numerics · p. 30
  • Finite-size states occur near −2 outside the infinite-layout spectrum: they are mainly edge-like in non-bipartite full-wave models and more bulk-like in half-wave and bipartite models.The full-wave non-bipartite states do not extend all the way to −2, leaving a finite gap; no corresponding high-energy interval was found within numerical resolution.Source: B. Numerics · p. 30, Figure 15 · p. 31
  • Middle gaps in finite-radius spectra close progressively with system size, and whether the limiting adjacency spectrum is one interval remains an open conjecture for k ≥ 7.The conjectured effective spectrum would have at most two intervals: the flat band and σ(A_Tk + 1).Source: B. Numerics · p. 30

VII. CONCLUSION

The conclusion connects circuit-QED line-graph lattices to graph operators and characterizes their flat bands and gaps. It also identifies hyperbolic finite-size states and gap-maximizing layouts.

VII. CONCLUSION

  • For CPW lattices with constant negative hopping, the effective spectrum is {−2}∞ ∪ σ(D_X ± A_X), with + for full-wave and − for half-wave modes.The full-wave Hamiltonian is equivalent to the line-graph Laplacian when hopping elements and on-site energies are equal.Source: §VII · p. 31
  • Both effective models have flat bands at −2 for any layout, whether finite or infinite, regular or irregular, homogeneous or inhomogeneous.The paper identifies adding photon-photon interactions to isolated flat bands as a starting point for quantum simulation of strongly correlated many-body physics.Source: §VII · pp. 31–32
  • For finite layouts, the full-wave model has a gap at −2 if and only if the layout is non-bipartite; half-wave gaps require an expander gap in the layout adjacency spectrum.For hyperbolic induced subgraphs, other cases have a macroscopic number of finite-size-induced states filling the infinite-layout ℓ2 gap.Source: §VII · p. 32
  • For degree-at-most-three layouts, the largest possible gap above the −2 flat band is (−2, −1), achieved by 3-regular Hoffman graphs.The paper also presents an infinite Cayley graph that is a universal cover of these examples and reports a separate gapped flat band at 0 for Hoffman layouts.Source: §VII · p. 32
  • For hyperbolic T7, the estimated infinite-layout lower gap is ∼0.41, while the truncated full-wave model displays a gap of ∼0.17.Finite-size-induced states near −2 are macroscopic; they are primarily whispering-gallery-like edge modes for non-bipartite full-wave models and mainly bulk-like otherwise.Source: §VII · p. 32

Appendix A: Construction of ¯Ha(X) from N and N t

The appendices relate line-graph operators to incidence operators, construct compactly supported eigenstates, and use reduced C∗-algebra results to bound spectral band counts.

Appendix A: Construction of H̄a(X) from N and N t

  • For half-wave modes, the oriented incidence operators give AX = DX − N^tN and H̄a(X) = NN^t − 2I.For full-wave modes, the edge adjacency operator is MM^t − 2I.Source: Appendix A · pp. 32–33

Appendix B: Compact Support Eigenstates

  • H̄s has a compactly supported eigenstate at −2 on every even cycle, while H̄a has one on every cycle.For H̄s, alternating amplitudes cancel at neighboring sites; for H̄a, a suitable orientation makes tunneling amplitudes cancel by destructive interference.Source: Appendix B · pp. 33–34

Appendix C: Reduced C∗-algebras

  • For the McLaughlin graph, the method gives at most 6 bands, whereas the graph has 4.The passage says these embedding-based bounds are not expected to be sharp without further understanding of the projections.Source: Appendix C · p. 34
  • For the tessellation graph Tk, the reduced C∗-algebra argument bounds the number of spectral bands by mk/3.The passage identifies mk as an arithmetic function of k; the McLaughlin graph example has at most 6 bands by this method, while its spectrum has 4.Source: Appendix C · pp. 34, 36

Appendix D: The Gap at −2 for ¯Hs(X) and finite X

Appendix D frames a lower-bound argument using edge cuts and the edge deletions needed to make induced subgraphs bipartite. Appendix E characterizes large cubic Hoffman layouts through line graphs.

Appendix D: The Gap at −2 for H̄s(X) and finite X

  • The Appendix D argument uses cut(S), the edges crossing from S to its complement, and emin(S), the minimum edge deletions needed to make the induced subgraph on S bipartite.It seeks a lower bound for λmin(DX + AX).Source: Appendix D · p. 36
  • For fixed r ≥ 2, the appendix applies its bound when every induced radius-r neighborhood is non-bipartite.The cited passage states this condition as the setting for the result, but does not include the resulting inequality.Source: Appendix D · p. 36

Appendix E: Classification of Hoffman Layouts

  • A large 3-regular Hoffman graph is a line graph of a subdivision graph of a 3-regular graph.The classification passage derives this form after excluding generalized line graph cases with nonzero auxiliary parameters.Source: Appendix E · p. 37

Appendix F: Subdivision Graphs and their Line Graphs

Appendix F relates a 3-regular graph, its subdivision graph, and the line graph of that subdivision. Block-matrix identities connect their characteristic polynomials.

Appendix F: Subdivision Graphs and their Line Graphs

  • Starting from a 3-regular graph X with 2ν vertices, its subdivision S(X) has 3ν degree-2 vertices and 2ν degree-3 vertices, and L(S(X)) has 6ν vertices.The subdivision graph has 6ν edges.Source: Appendix F · p. 38
  • The appendix uses block-matrix determinant identities and incidence operators to relate the characteristic polynomials of X, S(X), and L(S(X)).The passage describes this treatment as a summary of earlier proofs specialized to a degree-three starting graph.Source: Appendix F · p. 38
  • PL(S(X))(λ) = (λ + 2)^ν × λ^ν × PX(λ^2 − λ − 3).This is the stated polynomial relation for the line graph of the subdivision.Source: Appendix F · p. 38
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