Morse theory for discrete magnetic operators and nodal count distribution for graphs

Given a discrete Schr\"odinger operator $h$ on a finite connected graph $G$ of $n$ vertices, the nodal count $\phi(h,k)$ denotes the number of edges on which the $k$-th eigenvector changes sign. A {\em signing} $h'$ of $h$ is any real symmetric matrix constructed by changing the sign of some off-diagonal entries of $h$, and its nodal count is defined according to the signing. The set of signings of $h$ lie in a naturally defined torus $\mathbb{T}_h$ of ``magnetic perturbations"of $h$. G. Berkolaiko discovered that every signing $h'$ of $h$ is a critical point of every eigenvalue $\lambda_k:\mathbb{T}_h \to \mathbb{R}$, with Morse index equal to the nodal surplus. We add further Morse theoretic information to this result. We show if $h_{\alpha} \in \mathbb{T}_h$ is a critical point of $\lambda_k$ and the eigenvector vanishes at a single vertex $v$ of degree $d$, then the critical point lies in a nondegenerate critical submanifold of dimension $d+n-4$, closely related to the configuration space of a planar linkage. We compute its Morse index in terms of spectral data. The average nodal surplus distribution is the distribution of values of $\phi(h',k)-(k-1)$, averaged over all signings $h'$ of $h$. If all critical points correspond to simple eigenvalues with nowhere-vanishing eigenvectors, then the average nodal surplus distribution is binomial. In general, we conjecture that the nodal surplus distribution converges to a Gaussian in a CLT fashion as the first Betti number of $G$ goes to infinity.


Introduction
In some ways this paper is both an analog of [2] for discrete graphs and a continuation and expansion of the papers [7,16], although it is completely self-contained.

The Setting
Let be a simple graph on ordered vertices labeled 1, 2, • • • , .Write ∼ if ≠ are vertices connected by an edge.A (real or complex) function on is a function on the vertices of , that is, a vector in R or C and we denote the value of such a function = ( 1 , 2 , • • • , ) by ( ) or .An × matrix ℎ is supported on if ℎ ≠ 0 =⇒ ∼ or = .Let S( ) and H ( ) denote the vector spaces of real symmetric matrices and complex Hermitian matrices supported on .A discrete Schrödinger operator is a real symmetric matrix ℎ ∈ S( ) with ℎ < 0 for ∼ .The quadratic form associated with ℎ ∈ S( ) may be expressed as the quadratic form of Δ + , that is 2  (1.1) where the "potential" is ( ) = ℎ + ∼ ℎ and Δ is a weighted Laplace operator on .
A discrete Schrödinger operator ℎ has real eigenvalues 1 ≤ 2 ≤ • • • ≤ .Suppose is a simple (multiplicity one) eigenvalue of ℎ with a nowhere-vanishing eigenvector (meaning that ≠ 0 for all ).A basic problem in graph theory is to understand the behavior of the nodal count (ℎ, ), that is, the number of edges ∼ for which changes sign: ( ) ( ) < 0. It is known that where is the first Betti number of .(See [18] for a review of the many works leading to the upper bound, an analogue of Courant's theorem , and [12] for the lower bound.)This motivates the definition of the nodal surplus and its probability distribution (ℎ) = ( (ℎ) 0 , . . ., (ℎ) ) over the possible eigenvalues: In numerical simulations for large graphs, this distribution seems to concentrate around 2 with variance of the order of , similar to the observations for metric graphs in [2].

Nodal count for signed graphs
If ℎ ∈ S( ) is a discrete Schrödinger operator we may consider other signings ℎ ′ ∈ S( ) obtained from ℎ by changing the sign of some collection of off-diagonal entries.Every symmetric matrix ℎ ′ ∈ S( ) is a signing of a uniquely determined Schrödinger The Courant theorem states, for a domain Ω in Euclidean space with homogeneous boundary conditions, that the nodal set of the -th eigenfunction of the Laplacian divides Ω into no more than subdomains, see [17] Chapt.6 §6.operator ℎ.We may consider ℎ ′ to be an analog of the discrete Schrödinger operator on the corresponding signed graph ′ obtained from by attaching signs to the edges, as originally introduced in [24] and extensively studied, see [13,32,39].In this case, taking the signing into account, the nodal count is defined to be the number of edges ∼ such that ( )ℎ ′ ( ) > 0.
Denote by S(ℎ) the collection of all possible signings of ℎ (cf.§2.6).The inequality (1.2) continues to hold for any signing of ℎ.The average nodal surplus distribution (S(ℎ)) is the average of (ℎ ′ ) over all signings ℎ ′ ∈ S(ℎ).In Theorem 3.2 we show that if the diagonal entries of ℎ are all equal, then (S(ℎ)) is symmetric around /2. Numerical experiments lead to the following Conjecture.Given a simple connected graph there is a generic set (open, dense and full measure) of ℎ ∈ S( ) for which the average nodal surplus distribution (S(ℎ)) is symmetric around /2 with variance 2 ℎ of order .Moreover, the normalized distribution converges in the weak topology to the normal Gaussian distribution (0, 1) as → ∞, uniformly over all simple connected with first Betti number , and generic ℎ ∈ S( ).

Gauge invariance
The gauge group T = (R/2 Z) acts on the space H ( ) where ( 1 , 2 , • • • , ) acts by conjugation with diag( 1 , 2 , • • • , ).This action preserves eigenvalues, nodal count, and most other graph properties that are studied in this paper.Elements ℎ, ℎ ′ ∈ H ( ) that differ by a gauge transformation are said to be gauge equivalent.If ℎ ∈ S( ) is a discrete Schrödinger operator then the signings ℎ ′ of ℎ for which the corresponding signed graph ′ is balanced (see [24]) are exactly those ℎ ′ that are gauge equivalent to ℎ.

Magnetic operators and nodal count
In [7,9] G. Berkolaiko suggested that one might better understand the nodal count by considering its variation under magnetic perturbations of h.The discrete analog for the Schrödinger operator associated to a particle in a magnetic field appears in [25,26].See also [30], [14], [15, §2.1] and [16].It is quickly reviewed in Appendix B.
Given a discrete Schrödinger operator ℎ ∈ S( ), a magnetic potential is a real anti-symmetric matrix supported on and the associated magnetic Schrödinger operator ℎ ∈ H ( ) is the Hermitian matrix (ℎ ) = ℎ .The manifold (2.7) of such magnetic perturbations, T ℎ ⊂ H , is a torus containing ℎ, cf.§2.4 below.Its quotient, see equation (2.10), modulo gauge transformations, M ℎ is a torus of dimension .In [16] and [7], G. Berkolaiko and Y. Colin de Verdière discovered a remarkable fact: for any real symmetric ℎ ∈ S( ) with simple eigenvalue and nowhere vanishing eigenvector, the nodal surplus (ℎ, ) − ( − 1) is equal to the Morse index of , interpreted as a Morse function on the manifold M ℎ .

Morse theory for magnetic perturbations modulo gauge transformations
We wish to apply Morse theory to the function : H → R, restricted to the torus T ℎ or its quotient M ℎ .In principle, Morse theory provides a prescription for building the homology of M ℎ from local data at the critical points of together with some homological information as to how these local data fit together.Since the homology of M ℎ is known, Morse theory should provide restrictions on the number and type of critical points of , and in turn, restrictions on the nodal surplus.
There are several difficulties with this plan, the first being that is continuous but not smooth: it is analytic on each stratum of a certain stratification of H (see §7) [29,33].If (ℎ) is simple then is analytic near ℎ and one may search for its critical points on T ℎ .The torus T ℎ and its quotient M ℎ are preserved under complex conjugation, and the function is invariant under complex conjugation.The simplest critical points of are the symmetry points ( §2.6): the points ℎ ′ ∈ T ℎ (or [ℎ ′ ] ∈ M ℎ ) fixed by complex conjugation, i.e. the real symmetric matrices in T ℎ .
The set of symmetry points of T ℎ is denoted S(ℎ).If ℎ is real symmetric then S(ℎ) consists precisely of the various signings of ℎ.Following [6], we show: Theorem 3.2.Each critical point ℎ ′ ∈ T ℎ with simple eigenvalue (ℎ ′ ) and nowhere vanishing eigenvector is necessarily in the gauge equivalence class of a symmetry point.In other words, its image [ℎ ′ ] ∈ M ℎ is a symmetry point.Suppose that for each (1 ≤ ≤ ) each critical point ℎ ∈ T ℎ of has (ℎ ) as a simple eigenvalue with nowhere vanishing eigenvector.Then the average nodal count distribution is a binomial distribution with mean /2 and variance /4.Consequently, if the average nodal distribution is not binomial then there must exist critical points (of some eigenvalue) that are not symmetry points.
We give a homological characterization of symmetry points: Theorem 2.7.Let ℎ ∈ S( ) and ∈ A ( ) which we may identify as a 1-form on .

Classification of critical gauge-equivalence classes
In general, the nodal surplus distribution (S(ℎ)) depends on Morse data from all critical points of (for all ), whether or not they are symmetry points.Following Theorem 3.2, there are two possible types of non-symmetry critical points [ℎ ′ ] ∈ M ℎ of : (1) exceptional critical points, for which (ℎ ′ ) is simple but its eigenvector vanishes on one or more vertices.In this case [ℎ ′ ] is (usually) a degenerate critical point (see Theorem 4.4): it is contained in a larger critical submanifold.
(2) incorrigible critical points, for which the multiplicity of (ℎ ′ ) is greater than one.In this case, fails to be smooth and one must replace the usual Morse theory with stratified Morse theory ( [21]).
Concerning the first case, suppose the eigenvector vanishes only at a single vertex 0 of the graph .Suppose that 0 has degree deg( 0 ).Theorem 4.4.Assuming the critical point [ℎ ′ ] ∈ M ℎ is sufficiently generic then it lies in a nondegenerate (Morse-Bott) critical submanifold of M ℎ , of dimension deg( 0 ) − 3, which is diffeomorphic to the configuration space of a particular planar linkage.Its Morse index may be expressed in terms of spectral data.
The configuration spaces of planar linkages are fascinating objects.They have been extensively studied and their homology is completely known, cf.[19,27,37].
For the second case, when the multiplicity of (ℎ ′ ) is greater than one, G. Berkolaiko and I. Zelenko [11] have determined the normal Morse data for , and its Betti numbers, which forms the central ingredient required for stratified Morse theory.However, in order to apply stratified Morse theory to the mapping : T ℎ → R it is required that the manifold T ℎ ⊂ H should be Whitney stratified.Its stratification comes by intersecting with the natural stratification of H (cf. §7), but this requires that T ℎ should be transverse to the strata of the stratification of H .The challenge is to guarantee transversality of the torus T ℎ by a generic choice of the single element ℎ ∈ S( ).The transversality lemma in [11] does not address this situation.The first nontrivial case concerns the stratum 2 ( ) where has multiplicity 2. Suppose ℎ is a critical point of , an eigenvalue of multiplicity 2. In §7.8 we define the notion of a splitting of the graph by the eigenspace of .(A related condition was considered by L. Lovàsz in [31, §10.5.2].)Theorem 7.9.If the eigenspace of (ℎ ) does not split the graph then the space H ( ) is transverse to 2 ( ) at given point ℎ ∈ T ℎ .Specific conditions on ℎ ′ are given in Theorem 4.4 in §4.3 Corollary 7.10.As above, if the eigenspace of (ℎ ) does not split then for generic choice ℎ ′ ∈ S( ) the torus T ℎ ′ is transverse to the stratum 2 ( ) near ℎ .
Acknowledgements.The authors would like to thank Gregory Berkolaiko for enlightening discussions and for his comments on earlier versions of this paper.The first author would like to thank Nikhil Srivastava and Theo McKenzie for useful discussions.The idea of Theorem 3.2 (7) originated in joint discussions with Ram Band and Gregory Berkolaiko regarding metric graphs.The authors are very grateful to an anonymous referee for many thoughtful comments, which have considerably improved the paper.The first author would like to thank the Institute for Advanced Study, as this work began when he was a member there.
Funding.The first author was supported by the Ambrose Monell Foundation and the Simons Foundation Grant 601948, DJ.

Symmetric and Hermitian forms
Let S denote the vector space of × real symmetric matrices, A the space of × real antisymmetric matrices and H the space of × Hermitian matrices, that is, matrices of linear operators on C expressed in the standard basis and that are selfadjoint with respect to the standard Hermitian form , = ¯ .
If ⊂ C is a complex subspace then the standard Hermitian form restricts to a Hermitian form on and we denote by H ( ) the self adjoint linear operators → .If ∈ H then it may fail to preserve however its "restriction" to may be defined by expressing = * with respect to the decomposition C = ⊕ ⊥ .The restriction | is defined to be the operator ∈ H ( ).Equvalently, | is the operator corresponding to the restriction to , ∈ of the sesquilinear form ( , ) = , .

Laplace and Schrödinger operators
Throughout this section we fix a graph = ([ ], ).The natural ordering on the set of vertices [ ] := {1, 2, . . ., } determines an orientation for each edge.Write ∼ if ≠ and vertices , are joined by an edge.Write ≃ if ∼ or = .
A (real or complex) matrix supported on is an × matrix ℎ such that ℎ ≠ 0 =⇒ ≃ .Such a matrix is properly supported on if, in addition, ∼ =⇒ ℎ ≠ 0. Symmetric, antisymmetric and Hermitian matrices supported on are denoted S( ), A ( ), H ( ) respectively.Examples of matrices in S( ) include the adjacency matrix for , (weighted) Laplace operators for and discrete Schrödinger operators, see §1.1 above.More generally, any matrix ℎ ∈ H ( ) may be considered a magnetic Schrödinger operator for , (see §2.4 below and references [15,16]).

Graph homology
The space 0 ( ; Z) Z of 0-chains is the vector space of formal linear combinations of vertices, =1 [ ].Each edge with < is orientated from to so that the group 1 ( ; Z) of 1-chains is the group of formal linear combinations Then 1 ( ; Z) = ker( ) where : where is the number of connected components of .The vector space R may be viewed as the space of real-valued functions Ω 0 ( ) on the vertices of .If = ( 1 , 2 , • • • , ) we sometimes write = ( ).The vector space A ( ) of real, antisymmetric matrices supported on may be viewed as the space of 1-forms Ω 1 ( ) on with coboundary differential There are no 2-forms on a graph so 1 ( ; R) = Ω 1 ( )/ Ω 0 ( ) is canonically dual to the homology 1 ( ; R) under the the natural pairing that is determined by integration

Action of A
The vector space A = A (R) of × real antisymmetric matrices acts on the vector space H of × Hermitian matrices by for all ∈ A (R) and ℎ ∈ H with ( + ) * ℎ = * ( * ℎ) and with 0 * ℎ = ℎ.Then A ( ) acts on H ( ).
If ℎ is the discrete Schrödinger operator then * ℎ may be interpreted as the corresponding magnetic Schroödinger operator in the presence of a magnetic field described by , whose flux through a cycle is ∫ , with a sesquilinear form

Gauge invariance
The * action factors through the torus A (R)/A (2 Z).So the subtorus supported on T( acts on H ( ) by the * action.The differential (2.2) also factors through the gauge group T = R /(2 Z) .Gauge invariance is the statement that the * action by coboundaries is simply given by conjugation: for any ∈ T and any ℎ ∈ H , direct calculation gives The * action by preserves eigenvalues and preserves eigenvectors up to phase: Elements ℎ, ℎ ′ ∈ H ( ) that differ by a gauge transformation (ℎ ′ = * ℎ) are said to be gauge equivalent.Gague equivalence determines an identification (cf.(2.11) of the quotient torus (the manifold of magnetic fields modulo gauge transformations) with cohomology: T A/ ( ) := T( )/ (T ) 1 ( ; R/2 Z).

The embedded torus and its symmetry points
Recall (2.2) that a matrix ℎ ∈ H ( ) is properly supported on if ℎ ≠ 0 whenever ∼ .(Diagonal entries ℎ may vanish.)Such ℎ defines a mapping T( ) → H by ↦ → * ℎ, whose image is an embedding of T( ) into H ( ), We refer to T ℎ as the embedded torus.For ℎ ∈ H ( ) which is not properly supported on , the dimension of the embedded torus T ℎ is the number of nonzero elements ℎ with < .The embedded torus is invariant under complex conjugation and we refer to the set of its fixed points (i.e. the real points) as symmetry points.If ℎ ∈ S then its symmetry points S(ℎ) consist of symmetric matrices ℎ ′ obtained from ℎ by changing the signs in any subset of off-diagonal entries ℎ or equivalently ℎ ′ = * ℎ where ≡ 0 (m ). ( The action of the integral gauge group ( Z) ⊂ R preserves the set of symmetry points and changes the signs of the components of the corresponding eigenvectors.The set S(ℎ) decomposes into a union of orbits under the integral gauge group.If ℎ is properly supported on (ℎ ≠ 0 whenever ∼ ) then S(ℎ) has 2 | | elements, partitioned into 2 orbits (cf.§2.8).Each orbit corresponds to a choice of parity of the circulations around a choice of elementary cycles.
Proof.Since ℎ is properly supported, the element is uniquely determined modulo 2 Z.If ℎ is a symmetry point then ≡ 0 (m ) so (2.9) holds.If ℎ changes by gauge-equivalence, the integral (2.9) is unchanged, by Stokes' theorem.

Eigenvalues as Morse functions
Eigenvalues of elements ℎ ∈ H are real and ordered, say For each (1 ≤ ≤ ) the mapping : H → R is well defined, continuous and piecewise real-analytic: there is a stratification of H by analytic subvarieties such that the restriction of to each stratum is analytic (cf.§7.1 and Lemma 7.3).The restriction of each to the embedded torus T ℎ is invariant under gauge transformations so it determines a function on the quotient, where we use the notation / /T to denote dividing by gauge equivalence.The torus M ℎ has dimension , and is referred to in [16] as the manifold of magnetic perturbations modulo gauge transformations.
If ∈ R then ( ) * ℎ = (− ) * h so complex conjugation passes to an involution on M ℎ .Every fixed point of this involution comes from a symmetry point in T ℎ : for if ℎ ∈ H and [ℎ] ∈ M ℎ is fixed, this means h = ( ) * ℎ for some ∈ R , so ( 2 ) * ℎ is a symmetry point.It is therefore reasonable to refer to these fixed points of M ℎ as symmetry points of M ℎ .
2.9 Lemma.Let be a simple graph with connected components and let ℎ ∈ H ( ), properly supported on .Then, each symmetry point ℎ ′ ∈ S(ℎ) has exactly 2 − gauge-equivalent symmetry points.Thus, the number of symmetry points in Proof.It is enough to consider the case of real symmetric ℎ ∈ S , in which case its gauge-equivalent symmetry points are There are 2 choices for among which 2 are in the kernel of (those which are constant on connected components of ).So there are 2 − distinct values for , and therefore 2 − distinct values of * ℎ since ℎ is properly supported on .Hence, [ℎ] contains exactly 2 − gauge-equivalent symmetry points.Repeating this argument for any other ℎ ′ ∈ (ℎ) leaves 2 | | − ( − ) = 2 equivalence classes of symmetry points in M ℎ .

Nodal surplus
Generalizing the notions described in the introduction, let ℎ be a Hermitian matrix supported on , suppose (ℎ) is a simple eigenvalue with nowhere vanishing eigenvector Further assume that ¯ ℎ ∈ R for all ∼ (which is equivalent to ℎ being a critical point of , see Theorem 3.2 part (3)).Define the nodal count (ℎ, ) to be the number of edges ∼ such that ¯ ℎ > 0. (2.12) The nodal surplus is the number (ℎ, ) − ( − 1).This number does not change under gauge transformation and it is known (see Theorem 3.2 below) that the nodal surplus is between 0 and , the first Betti number of .The nodal surplus distribution (ℎ) = (ℎ) 0 , (ℎ) 1 , . . ., (ℎ) is the vector representing the probability distribution of these numbers over the possible eigenvalues: Assuming that ℎ ∈ S and all its signings ℎ ′ ∈ S(ℎ) have all eigenvalues simple with nowhere-vanishing eigenvectors, the distribution can be averaged over signings to give the average nodal distribution

Critical points
Throughout this section we fix a graph with vertices 1, • • • , and edges ∼ .Let ℎ ∈ S( ) be a real symmetric matrix properly supported on , cf. §2.2.For ∈ A ( ) denote by ℎ = * ℎ the magnetic perturbation of ℎ.Fix and write ( ) = (ℎ ) for the k-th eigenvalue.Let M ℎ be the manifold (2.10) of magnetic perturbations of ℎ modulo gauge transformations.It is a torus of dimension , the first Betti number of the graphs .By equation (2.6) the eigenvalue ( ) of an element [ℎ ] ∈ M ℎ , and its multiplicity are well defined; and whether or not an eigenvector vanishes at a given vertex is well defined.We consider : M ℎ → R to be a sort of generalized Morse function.If is smooth at a point = [ℎ ] ∈ M ℎ (in which case it is also analytic) we say that is a smooth point of .A critical point of is either a non-smooth point or a smooth point where ∇ ( ) = 0. Consider the following possibilities: (0) may be a smooth, regular (i.e., not critical) point of .
(1) may be a symmetry point of M ℎ .
(3) may be a non-smooth point of .
(1) Every symmetry point of M ℎ is a critical point of .
(2) If the only critical points of on M ℎ are the symmetry points and if they are nondegenerate then the number of such critical points of index is .
(3) Suppose ℎ ∈ T ℎ has a simple eigenvalue (ℎ ) with eigenvector .Then (ℎ ) ¯ is real for all ∼ if and only if ℎ is a critical point of as a function on T ℎ , in which case ℎ is gauge equivalent to a matrix ℎ ′ such that ℎ ′ ∉ R =⇒ ¯ = 0.
(6) If the diagonal entries of ℎ are all equal then the average nodal count distribution is symmetric, (7) Suppose that for each (1 ≤ ≤ ) each critical point ℎ ∈ T ℎ of has (ℎ ) as a simple eigenvalue with nowhere vanishing eigenvector.Then the average nodal count distribution is binomial: Parts (1) and ( 5) of Theorem 3.2 are due to Berkolaiko and Colin de Verdière [7,16].Part ( 2) is an immediate consequence, also known to both of these authors.Part (3) was already observed in [6, Theorem A.1 and Lemma A.2 ].Part ( 4) is an immediate consequence known to the authors of [6].It says that the only simple critical points of |T ℎ with non-vanishing eigenvector occur along the intersection of T ℎ with the conjugacy classes of symmetry points: the 2 | | real elements S(ℎ).Proofs for Theorems 3.2 and 4.4 below will appear in §5 and §6.

Exceptional critical points and the linkage equation
In this section we consider the case where [ℎ ] ∈ M ℎ is an exceptional critical point of (cf.§1.6).That is, [ℎ ] is a non-symmetry, smooth, critical point with (ℎ ) simple.According to Theorem 3.2 the eigenvector corresponding to (ℎ ) vanishes somewhere.(By generic choice of ℎ we can guarantee that every eigenvector of ℎ is nowhere vanishing (cf.[36]) but we cannot guarantee the same holds for all ℎ ∈ T ℎ .)We address the simple case of eigenvector that vanishes at a single vertex.By possibly replacing ℎ with a gauge equivalent ℎ + and with we may assume that is real with non-negative entries.The setting for Theorem 4.4 is described next.
In both [7] and [16] and the matrix ℎ was assumed to be real symmetric but essentially the same proof works in general.

Configuration space of a planar linkage
Equation (4.1) implies that the following planar linkage equation ([19, 27, 37]) holds: This equation (4.2) describes a collection of vectors ∈ C = R 2 in the plane, placed end to tail, that starts and ends at the origin, that is, a planar linkage, depending on a collection of lengths = { } ∈ 0 .Let 1 ⊂ C be the unit circle.The configuration space Θ (see [19]) of the planar linkage defined by (4.2) is the set of solutions modulo rotations, that is, where the unit circle acts diagonally on ( 1 ) 0 by multiplication.The planar linkage is said to be generic if for any ∈ {−1, 1} 0 , Let be the maximal length, = max( ) ∈ 0 .If > ≠ then there are no solutions, Θ = ∅.If the planar linkage is generic and 27,37]) whose Betti numbers have been computed in [19,23].Let be the second largest length.If + ≤ 1 2 then Θ is connected, otherwise it has two connected components, exchanged by complex conjugation, each diffeomorphic to the torus of dimension | 0 | − 3.

Exceptional points
In the notation of §4.1, suppose ℎ = * ℎ is an exceptional critical point of with real eigenvector = (0, ′ ) and simple eigenvalue = (ℎ ).The complex conjugate point h = (− ) * ℎ is also a critical point of , with the same eigenvalue and eigenvector .Moreover, is also an eigenvalue of , say = ′ ( ) is its ′ -th eigenvalue.
Let be the connected component of the critical set in M ℎ of that contains ℎ , union with the connected component of the critical set of that contains h , noting that these two sets may be the same .
Then the critical set coincides with the explicitly defined set Thus, the set ⊂ M ℎ has either one or two connected components.
A related observation for periodic metric (quantum) graphs appears in [10] §3.4,where certain graphs are constructed so that the maximum of their first spectral band is obtained on a critical manifold which is a planar linkage configuration space.

Proof of Theorem 3.2 5.1.
For part (1), suppose ℎ ′ ∈ T ℎ is a symmetry point (of T ℎ ), namely ℎ ′ = h′ ∈ S(ℎ).If ℎ ′ is not a smooth point of , then it is a critical point.Suppose ℎ ′ is smooth, then the directional derivative of in the direction ∈ A ( ) is then it is conjugate to ℎ ′ .Conjugation takes a neighborhood of ℎ ′ in H ( ) to a neighborhood of ℎ ′′ , preserving the eigenvalue so it also preserves the derivative of .
Part (2) follows immediately from the Morse inequalities, (M ℎ ) ≥ (M ℎ ) where denotes the number of critical points of index and where is the -th Betti number of M ℎ .There are 2 criticial points by Lemma 2.9, and the sum of the Betti numbers of M ℎ , a -dimensional torus, is also 2 .So = = for all .Assuming parts (4) and ( 5), the proof of part ( 7) is a simple computation.In Part (7) we assume all the critical points of correspond to simple eigenvalues with nowherevanishing eigenvectors, which means that all critical points are nondegenerate and are symmetry points, by Part (4).Part (5) says that in such cases the nodal surplus equals the Morse index.Therefore, the average nodal surplus is Using Lemma 2.9 the number inside the parenthesis can be expressed on the quotient because, by part (2), the number in the parentheses is independent of .For part (6), by subtracting a multiple of the identity we may assume the diagonal entries of ℎ are all zero.Let ∈ A ( ) be properly supported on , with ± on the non-zero entries.For any ℎ = * ℎ ∈ T ℎ the element −ℎ = ( + ) * ℎ ∈ T ℎ is also in the same torus but the order of the eigenvalues is reversed, (ℎ ) = − (−ℎ ).This results in an inversion that sends every critical point of with index , to a critical point of − with index − .When averaged it gives the needed symmetry around /2.

5.2.
In this paragraph we prove parts (3) and ( 4) of Theorem 3.2.Let h ∈ T ℎ and suppose that ( h) has multiplicity one.is analytic in a T ℎ neighborhood of h, and we ask when is it a critical point.To ease notation, for this paragraph only, we replace h by ℎ so that ℎ is now Hermitian rather than real symmetric.Fix a direction ∈ 0 (T( )) = A ( ) and consider the one-parameter perturbation of ℎ in that direction ℎ = ( 0 ) * ℎ for small ∈ (− , ) so that ℎ = ℎ .Since is simple, we get analytic functions ( ) ∈ C and ( ) ∈ C, such that for all ∈ (− , ), the vector ( ) is normalized and satisfies ℎ ( ) = ( ) ( ).Using Leibniz "dot" notation for derivative with respect to at = 0 we have ℎ + ℎ = + . (5.1) Taking the inner product with = (0), using that ℎ is self adjoint, gives In particular, if is nowhere vanishing then ℎ ′ is a symmetry point.If ℎ ′ ∉ R then either = 0 or = 0.This completes the proof.

5.3.
In the following paragraphs we prove part (5) of Theorem 3.2.The result was proven by Berkolaiko [7] and Colin de Verdière [16] for real symmetric ℎ with non-positive off-diagonal entries.Both proofs extend to any real symmetric ℎ, as the authors noted, if one defines the nodal count as in equation (2.12).We reorganize the proof of [16] and present it here for completeness and for later use.Now let ℎ ∈ T ℎ ⊂ H ( ) be an element whose equivalence class is a symmetry point, i.e., ℎ is gauge equivalent to a real symmetric matrix.For convenience we change the notation slightly, using ℎ instead of ℎ , so suppose ℎ ∈ H ( ) is Hermitian properly supported on , which is a critical point of , with a simple eigenvalue := (ℎ) and a nowhere-vanishing eigenvector .By Theorem 3.2,

Step 2
For (small) ∈ R and ∈ R we will find the second derivative of

Step 3
For any ∈ R , the derivative of in direction is ′ = Θ and Hess( ) = 0 due to gauge invariance.Let ℎ stand for the derivative of ℎ in direction ∈ A ( ), so that for any = equation (5.5) gives , Hess( ) It follows from (5.2) that Θ , ℎ is purely imaginary, so ∈ if and only if Θ , ℎ vanish for all real diagonal Θ.Since is nowhere-vanishing, Consequently, equation (5.5) shows that Hess( ) and Hess( ) agree on .

Proof of Theorem 4.4 6.1. The critical set ′
Recalling the notations of §4.1, the graph has + 1 vertices labeled 0, 1, . . ., .The set 0 is the set of edges connected to 0. The graph is the induced graph on the non-zero vertices.The torus of perturbations and its tangent space decompose: and ℎ = * ℎ = * is an exceptional critical point of with simple eigenvalue = (ℎ ) and real eigenvector = (0, ′ ) with 0 = 0 and > 0 for > 0. As discussed in §4.1, must be real, so it is a signing of .By replacing ℎ ∈ S( ) with a signing of ℎ (if necessary) we may assume = and ∈ A ( 0 ), so ℎ = becomes ′ = ′ and Recall that denotes the union of the connected components of the critical set of in M ℎ that contain [ℎ ] and [ h ].In §6.2 and §6.8 we will prove that = ′ where Observe that the set ′ is closed under complex conjugation because and are real, and * ℎ = (− − 2 ) * ℎ for any ∈ A ( 0 ).Moreover, the set ′ consists of critical points of : since ℎ is critical and is simple, Theorem 3.2 part (3) implies (ℎ ) is real for every ∼ .Since = (ℎ ′ ) is simple for any ℎ ′ = * ℎ ∈ ′ , then ℎ ′ = (ℎ ) is real for every ∼ (since 0 = 0) hence [ℎ ′ ] is also a critical point.
(The reverse inclusion is proven in §6.8.)

Gauge transformations on , and 0
Decompose the space of functions on , 2) The image R is the projection of R into A ( ) and in fact, for any ∈ R , where ( , 0) ∈ A ( 0 ) is the antisymmetric matrix with ( , 0) 0 = 1, ( , 0) 0 = −1 and all other entries are 0. Let 1 = (0, 1, 1, . . ., 1) ∈ R denote the constant vector on and let Since is properly supported on and = ′ ( ) is a simple eigenvalue of with a nowhere-vanishing eigenvector ′ , then is connected.Theorem 3.2 gives a decomposition and can be described in terms of directional derivatives, according to §5.6, 2).
Proof.It follows from (6.3) that R ⊂ R +1 + A ( 0 ).Using (6.4) and (6.5), so A ( ) is spanned by the sum on the left side.On the other hand, the sum on the left hand side is a direct sum because the sum of the dimensions of the vector spaces is

The tangent space to ′
Consider the preimage of ′ T( 0 ) * ℎ , Differentiate and use the identification ℎ T ℎ = A ( ) to obtain the tangent space ℎ ′ ∈ A ( 0 ) : By Lemma 6.4 the quotient projection takes Let R 0 0 be the space of mean zero elements of R 0 , which we identify with A 0 ( 0 ).Let x ∈ R 0 0 and y ∈ R 0 0 such that ( ) = x + y for all ∈ 0 .Then

is Morse-Bott
Recall that the submanifold of critical points 7. Transversality to the strata of H

The strata
The vector space H of Hermitian × matrices is stratified according to the multiplicities of the eigenvalues, as described in [5].(See also [3,4,34].)Suppose ℎ ∈ H has distinct eigenvalues 1 < 2 < • • • < .Specifying a multiplicity ( ) for the eigenvalue determines a stratum ( ), consisting of Hermitian matrices with eigenvalues and multiplicities ( ).The multiplicity vector is an ordered partition of , meaning that = =1 ( ), and every ordered partition of determines a stratum.The set of possible eigenvalues for ℎ forms an open set Therefore the stratum ( ) may be canonically identified with the product where F ( ) denotes the partial flag manifold of subspaces 0 This identification endows the stratum ( ) with the canonical structure of an analytic manifold, and each eigenvalue : ( ) → R is an analytic function.
where is the bundle whose fiber over ( , ) is the Grassmannian of − 1 dimensional complex subspaces − ⊂ ⊥ .From this we see that ( ) is an analytic manifold and = is an analytic function on ( ), as a coordinate in the parametrization.The dimension of ( ) may be calculated from the above diagram, dim and a miraculous cancellation of terms gives codim( ( )) = 2 − 1.It is a direct consequence of the definitions that ( ) is the intersection (7.3).The intersection is transversal because the different factors in (7.3) involve independent conditions.Proof.Let be a tangent vector to ( , ) at the point ℎ = ℎ 0 .Let ℎ ∈ ( , ) be a smooth one parameter family with = ℎ = ℎ(0).Suppose ∈ ⊂ C is an eigenvector of ℎ with eigenvalue .Differentiating the eigenvalue equation ℎ = gives ℎ + ℎ = .Taking the inner product with any ∈ gives , = 0 which shows that has the form of equation (7.4) above.On the other hand the codimension of the space of matrices (7.4) is 2 which equals the codimension of ( , ) so (7.4) describes the full tangent space.A similar procedure works for the tanget space to ( ).Part (B) of the proposition is an immediate consequence.
For part (C), using the decomposition C = ⊕ ⊥ , the matrix of = ℎ − . is where is nonsingular.Given ∈ × (C), we have In fact it is multiplication by the directional derivative ( ).
A closely related result, concerning transversality with respect to the manifold of matrices with constant rank, appears in [31,Chapter 10.5].Although Lovász considers only real symmetric matrices, his proof applies also to Hermitian matrices.

Application to graphs
Fix , , ≥ 1 and let ∈ R. Recall ( ), ( , ) ⊂ H from (7.1), (7.2).Let be a graph on vertices with a set of edges and associated spaces S( ) ⊂ H ( ) ⊂ H .In this section we determine when the inclusion H ( ) → H is transverse to the manifold ( ) at a point ℎ of their intersection.The results are used in Corollary 7.10, the case of multiplicity 2, to provide sufficient conditions which guarantee that the mapping T ℎ → H is transverse to 2 ( ) locally near ℎ.
Let H ( ) denote those Hermitian matrices that are supported on the complement of .That is ℎ ∈ H ( ) if ℎ * = ℎ, ℎ = 0, and ℎ = 0 for all and for any edge .It is the orthogonal complement in H to H ( ).

Example -Graphs for which T is generically transverse to 2
Suppose is a graph obtained by removing a set of disjoint edges from the complete graph, say ( , ) for = 1, . . ., such that the vertices { 1 , 1 , 2 , 2 , . ..} are all distinct.If ℎ ∈ H ( ) has distinct diagonal elements (a generic assumption) then the embedded torus T ℎ intersects 2 ( ) transversally for every .To prove this, it suffices by Corollary 7.10 to show, for any ℎ ∈ T ℎ , that no multiplicity-two eigenvalue of ℎ splits .
Assume by contradiction that some ℎ ∈ T ℎ has a multiplicity two eigenvalue = (ℎ ) with eigenspace such that the induced graph |spt( ) is disconnected.By the construction of this means that spt( ) = { , } for one of the missing edges ( , ).So is a multiplicity-two eigenvalue of the restriction This contradicts the assumption that diagonal elements of ℎ are distinct.

A. Transversality
A Remarks.Here, T is any finite-dimensional smooth manifold.The symbol T is being used to indicate that for our application, T is an open subset of the torus T( ).This result says, for example, that two submanifolds of Euclidean space may be made transverse by an arbitrarily small translation.The transversality lemma is due originally to R. Thom ([35]).The proof described here may be found in ( [22]).
Proof.It suffices to consider the case when is open in some Euclidean space.By assumption, the set = Φ −1 ( ) is a smooth submanifold of T × and it is easy to check that ∈ is a regular value of the projection : → if and only if the partial map : T → H is transverse to .But Sard's theorem says that the set of non-regular values of has Lebesgue measure zero.Now assume is closed and Φ is proper (i.e., the preimage of a compact set is compact).To show the set of "transversal" elements ∈ is open, we show its complement is closed.Let ∈ be a convergent sequence of points, say → ∈ for which there exists points ∈ T such that fails to take the tangent space T transversally to where = Φ( , ).Since Φ is proper, by taking a subsequence if necessary we may assume the sequence converge, say → ∈ T and therefore → for some ∈ H. Since is closed, we also have ∈ .The failure of transversality is a closed condition so fails to take T transversally to .Finally, if Φ, T, , H, are analytic then the set of points ( , ) ∈ T × for which fails to be transverse at is again analytic so its image ⊂ is a subanalytic subset of .It has positive codimension, for if contains an open set in then this contradicts the assumption that Φ is transverse to .

B. Heuristics for discretization of magnetic Schrödinger operators
The definition of discrete magnetic operators can be found in [14,30] for example, however, we will give here a heuristic explanation for why this is the right discretization for magnetic Schrödinger operators.For simplicity, we consider domains in R 3 so that magnetism can be described using vector fields: a magnetic field and magnetic potential such that = ∇ × .(The modern approach would consider and as a 1-form and 2-forms).The quadratic form of a Schrödinger operator = Δ + on a domain Ω ⊂ R is where Λ ⊂ Ω is a grid of side length .Introducing a magnetic field , the operator is changed to a magnetic Schrödinger operator by the rule ( ) ↦ → ( ) + ) instead of the quadratic form of ℎ in (1.1).If ℎ = (ℎ ) ∈ H define |ℎ| ∈ S by |ℎ| = |ℎ | for ≠ and |ℎ| = ℎ (diagonal entries of |ℎ| can be negative).Then there exists ∈ A so that ℎ = * (|ℎ|).

2 ,
for the relevant class of functions on Ω.If we approximate ( ) with ′ ⊂ M ℎ is a Morse-Bott critical submanifold of if at every point [ℎ ′ ] ∈ ′ , the kernel of Hess ([ℎ ′ ]) is exactly the tangent space [ℎ ′ ]′ and the number of negative eigenvalues of Hess ([ℎ ′ ]) is constant for all [ℎ ′ ] ∈ ′ .By (6.9) the kernel condition holds at [ℎ ].Since (ℎ ) is nonzero and continuous it does not change sign, so the Morse-Bott condition holds at every point[ℎ ′ ] ∈ ′ .As the kernel of the Hessian at a point[ℎ ′ ] ∈ ′ ⊂ is [ℎ ′ ]′ and is constant on , the tangent spaces agree,[ℎ ′ ] ′ = [ℎ ′ ] .Since ′ is closed it is aunion of connected components of .It contains both [ℎ ] and its complex conjugate, so = ′ is Morse-Bott and ind is transverse to .If Φ is proper and ⊂ H is closed then this set of values is open in.If Φ, T, , H and are analytic then the set of values ∈ for which transversality of fails is a subanalytic subset of of positive codimension.
.1 Transversality Lemma.Let Φ : T × → H be a smooth map between smooth manifolds and suppose this map is transverse to a submanifold ⊂ H. Then there is a dense set of values ∈ such that the partial map : T → H given by ( ) = Φ( , )