Nodal domains on graphs  How to count them and why?


 Elfreda Burns
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1 Proeedings of Symposia in Pure Mathematis Nodal domains on graphs  How to ount them and why? Ram Band, Idan Oren and Uzy Smilansky, Abstrat. The purpose of the present manusript is to ollet known results and present some new ones relating to nodal domains on graphs, with speial emphasize on nodal ounts. Several methods for ounting nodal domains will be presented, and their relevane as a tool in spetral analysis will be disussed. 1. Introdution Spetral graph theory deals with the spetrum and eigenfuntions of the Laplae operator defined on graphs. The study of the eigenfuntions, and, in partiular, their nodal domains is an exiting and rapidly developing researh diretion. It is an extension to graphs of the investigations of nodal domains on manifolds, whih started already in the 19th entury by the pioneering work of Chladni on the nodal strutures of vibrating plates. Counting nodal domains started with Sturm s osillations theorem whih states that a vibrating string is divided into exatly n nodal intervals by the zeros of its n th vibrational mode. In an attempt to generalize Sturm s theorem to manifolds in more than one dimension, Courant formulated his nodal domains theorem for vibrating membranes [1], whih bounds the number of nodal domains of the n th eigenfuntion by n. Pleijel [] have shown later that Courant s bound an be realized only for a rare subsequene of eigenfuntions. The study of nodal domains ounts was revived after Blum et al have shown that nodal ount statistis an be used as a riterion for quantum haos [3]. A subsequent paper by Bogomolny and Shmit [4] illuminated another fasinating onnetion between nodal statistis and perolation theory. A reent paper by Nazarov and Sodin addresses the ounting of nodal domains of eigenfuntions of the Laplaian on S [5]. They prove that on average, the number of nodal domains inreases linearly with n, and the variane about the mean is bounded. At the same time, it was shown that the nodal sequene  the sequene of numbers of nodal domains ordered by the orresponding spetral parameters  stores geometrial information about the domain [6]. Moreover, there is a growing body of numerial and theoretial evidene whih shows that the nodal sequene an be used to distinguish between isospetral manifolds. [7, 8]. In the present paper we shall fous on the study of nodal domains on graphs, and show to what extent it goes hand in hand or omplements the orresponding results obtained for Laplaians on manifolds (opyright holder)
2 RAM BAND, IDAN OREN AND UZY SMILANSKY, The paper is designed as follows: The next hapter summarizes some elementary definitions and bakground material neessary to keep this paper self ontained. Next, we survey the known results regarding ounting nodal domains on graphs. After these preliminaries, we present a few ounting methods of nodal domains on graphs. Finally, the intimate onnetion between nodal sequenes and isospetrality on graphs will be reviewed, and some open problems will be formulated.. Definitions, notations and bakground A graph G = (V, B) is a set of verties V = {1,, V } of size V V and a set of undireted bonds (edges) B of size B B, suh that (i,j) B if the verties i and j are onneted by a bond. In this ase we say that verties i and j are adjaent and denote this by i j. The degree (valeny) of a vertex is the number of bonds whih are onneted to it. A graph is alled vregular if all its verties are of degree v. Throughout the artile, and unless otherwise stated, we deal with onneted graphs with no multiple bonds or loops (a bond whih onnets a vertex to itself). A well known fat in graph theory is that the number of independent yles in a graph, denoted by r is equal to: (1) r = B V + Co where Co is the number of onneted omponents in G. We note that r is also the rank of the fundamental group of the graph. A tree is a graph for whih r = 0. Let g be a subgraph of G. We define the interior of g as the set of verties whose neighbors are also in g. The boundary of g is the set of verties in g whih are not in its interior. A graph G is said to be properly olored if eah vertex is olored so that adjaent verties have different olors. G is kolorable if it an be properly olored by k olors. The hromati number χ(g) is k if G is kolorable and not (k1)olorable. A very simple observation, whih we will use later, is that χ(g) V. G is alled bipartite if its hromati number is 1 or. However, sine a hromati number 1 orresponds to a graph with no bonds, and we are dealing only with onneted graphs, we an exlude this trivial ase and say that for a bipartite graph, χ =. The vertex set of a bipartite graph G an be partitioned into two disjoint sets, say V 1 and V, in suh a way that every bond of G onnets a vertex from V 1 with a vertex from V. We then have the following notation: G = (V 1 V, B) [0]. The adjaeny matrix (onnetivity) of G is the symmetri V V matrix C = C(G) whose entries are given by: { 1, if i and j are adjaent C ij = 0, otherwise Laplaians on graphs an be defined in various ways. The most elementary way relies only on the topology (onnetivity) of the graph, and the resulting Laplaian is an operator on a disrete and finitedimensional Hilbert spae. These operators or their generalizations to be introdued below will be referred to as Disrete or ombinatorial Laplaians. One an onstrut the Laplaian operator as a differential operator if the bonds are endowed with a metri, and appropriate boundary onditions are required at the verties. The resulting operator should be referred
3 NODAL DOMAINS ON GRAPHS  HOW TO COUNT THEM AND WHY? 3 to as the Metri Laplaian. However, beause the metri Laplaian is idential with the Shrödinger operator on the graph, one often refers to this system as a Quantum Graph  a misnomer whih is now hard to eradiate. In the sequel we shall properly define and disuss the relevant versions of Laplaians on graphs. The disrete Laplaian, of G, is the matrix () L(G) = D C, where D is the diagonal matrix whose i th diagonal entry is the degree of the vertex v i, and C is the adjaeny matrix of G. A generalized Laplaian, L is a symmetri V V matrix with offdiagonal elements defined by: L ij < 0 if verties i and j are adjaent, and L ij = 0 otherwise. There are no onstraints on the diagonal elements of L. The eigenvalues of L(G) together with their multipliities, are known as the spetrum of G. To eah eigenvalue orresponds (at least one) eigenvetor whose entries are labeled by the verties indexes. It is well known that the eigenvalues of the ombinatorial Laplaian are nonnegative. Zero is always an eigenvalue and its multipliity is equal to the number of onneted omponents of G. An important property regarding spetra of large vregular graphs is that the limiting spetral distribution is symmetri about λ = v, and is supported in the interval [v v 1,v + v 1] [18]. An extensive survey of the spetral theory of disrete Laplaians an be found in [1,, 3]. To define quantum graphs a metri is assoiated to G. That is, eah bond is assigned a positive length: L b (0, ). The oordinate along the bond b is denoted by x b. The total length of the graph will be denoted by L = b B L b. This enables to define the metri Laplaian (or Shrödinger) operator on the graph as the Laplaian in 1d d dx on eah bond. The domain of the Shrödinger operator on the graph is the spae of funtions whih onnet ontinuously aross verties and whih belong to the Sobolev spae H (b) on eah bond b. Moreover, vertex boundary onditions are imposed to render the operator self adjoint. We shall onsider in this paper the Neumann and Dirihlet boundary onditions: (3) Neumann ondition on the vertex i : ψ b (x b ) = 0, dx b (4) b S (i) d xb =0 Dirihlet ondition on the vertex i : ψ b (x b ) xb =0 = 0, where S (i) denotes the group of bonds whih emerge from the vertex i and the derivatives in (3) are direted out of the vertex i. The eigenfuntions are the solutions of the bond Shrödinger equations: (5) b B d dx ψ b = k ψ b, whih satisfy at eah vertex boundary onditions of the type (3) or (4). The spetrum {kn} n=1 is disrete, nonnegative and unbounded. One an generalize the Shrödinger operator by inluding potential and magneti flux that are defined on the bonds. Other forms of boundary onditions an also be used. However, these generalizations will not be addressed here, and the interested reader is referred to two reent reviews [13, 14].
4 4 RAM BAND, IDAN OREN AND UZY SMILANSKY, Finally, two graphs, G and H, are said to be isospetral if they posses the same spetrum (same eigenvalues with the same multipliities). This definition holds both for disrete and quantum graphs. 3. Nodal domains on graphs Nodal domains on graphs are defined differently for disrete and metri graphs. Disrete graphs: Let G = (V, B) be a graph and let f=(f 1,f,...,f V ) be a real vetor. We assoiate the real numbers f i to the verties of G with i = 1,,...,V. A nodal domain is a maximally onneted subgraph of G suh that all verties have the same sign with respet to f. The number of nodal domains with respet to a vetor f is alled a nodal domains ount, and will be denoted by ν(f). The maximal number of nodal domains whih an be ahieved by a graph G will be denoted by ν G. The nodal sequene of a graph is the number of nodal domains of eigenvetors of the Laplaian, arranged by inreasing eigenvalues. This sequene will be denoted by {ν n } V n=1. The definition of nodal domains should be sharpened if we allow zero entries in f. Two definitions are then natural: A strong positive (negative) nodal domain is a maximally onneted subgraph H of G suh that f i > 0 (f i < 0) for all i H. A weak positive (negative) nodal domain is a maximally onneted subgraph H of G suh that f i 0 (f i 0) for all i H. In both ases, a positive (negative) nodal domain must onsist of at least one positive (negative) vertex. Aording to these definitions, it is lear that the weak nodal domains ount is always smaller or equal to the strong one. Metri graphs: Nodal domains are onneted domains of the metri graph where the eigenfuntion has a onstant sign. The nodal domains of the eigenfuntions are of two types. The ones that are onfined to a single bond are rather trivial. Their length is exatly half a wavelength and their number is on average kl π. The nodal domains whih extend over several bonds emanating from a single vertex vary in length and their existene is the reason why ounting nodal domains on graphs is not a trivial task. The number of nodal domains of a ertain eigenfuntion on a general graph an be written as (6) ν n = 1 i b S (i) { k nl b π + 1 ( 1 ( 1) knl b π sign[φ i ]sign[φ j ]) } B + V where x stands for the largest integer whih is smaller than x, and φ i,φ j are the values of the eigenfuntion at the verties onneted by the bond b = {i,j} [17]. (6) holds for the ase of an eigenfuntion whih does not vanish on any vertex: i φ i 0, and there is no yle of the graph on whih the eigenfuntion has a onstant sign. The last requirement is true for high enough eigenvalues where half the wavelength is smaller than the length of the shortest bond. This restrition, whih is important for low eigenvalues, was not stated in [17]. Nodal domains on quantum graphs an be also defined and ounted in an alternative way. The values of the eigenfuntions on the verties assoiate the vetor φ = (φ 1,...,φ V ) to the vertex set, and its number of nodal domains is ounted as was explained above. The reasoning behind this way of ounting is that the values of the eigenfuntion on the verties {φ i } together with the eigenvalue k
5 NODAL DOMAINS ON GRAPHS  HOW TO COUNT THEM AND WHY? 5 store the omplete information about the values of the eigenfuntion everywhere on the graph. We thus have two independent ways to define and ount nodal domains. To distinguish between them we shall refer to the first as metri nodal domains, and the number of metri domains in the n th eigenfuntion will be denoted by µ n. The domains defined in terms of the values of the eigenfuntion on the verties will be referred to as the disrete nodal domains. The number of disrete nodal domains of the n th eigenfuntion will be denoted by ν n, similar to the notation of this ount on the disrete graphs. As far as ounting nodal domains is onerned, trees behave as one dimensional manifolds, and the analogue of Sturm s osillation theory applies for the eigenfuntions of the disrete [9] and the metri Laplaians [10, 11, 1, 15], as long as the eigenvetor (or the eigenfuntion) does not vanish at any vertex. Thus we have ν n = n for disrete tree graphs and µ n = n for metri ones. Similarly, Courant s theorem applies for the eigenfuntions of both the disrete and the metri versions of the Laplaian on any graph, [16, 17]. However, sharper lower and upper bounds for the number of nodal domains were disovered reently. Berkolaiko [7] provided a lower bound for the nodal domains ount for both the disrete and the metri ases. He showed that the nodal domains ount of the n th eigenfuntion of a Laplaian (either disrete or metri) has no less than n r nodal domains. Again, this is valid if the eigenfuntion has no zero entries and it belongs to a simple eigenvalue. When n r < 0, this result is trivial sine a nodal domains ount is positive by definition. We note that for metri graphs this theorem holds only when the metri ount is used. A global upper bound for the nodal domains ount of a graph G was derived in [8]: The maximal number of nodal domains on G was proven to be smaller or equal to V χ +, where χ is the hromati number of G. This bound is valid for any vetor, not only for Laplaian eigenvetors. To end this setion we shall formulate and prove a few results whih show that not all possible subgraphs an be nodal domains of eigenvetors of the disrete Laplaians. The topology and onnetivity of nodal domains are restrited, and the restritions depend on whether the eigenvalue is larger or smaller than the spetral midpoint v. Theorem 3.1. Let G be a vregular graph, and let f be an eigenvetor of the disrete Laplaian, orresponding to an eigenvalue λ. Let g be a nodal domain of f. Then the following statements hold: i. For all eigenvetors with eigenvalue λ > v the nodal domains do not have interior verties. ii. For all eigenvetors with eigenvalue λ < v, all the nodal domains onsist of at least two verties. iii. For all eigenvetors with eigenvalue λ < v k (and k < v), in every nodal domain there exists at least one vertex with a degree (valeny) whih is larger than k.
6 6 RAM BAND, IDAN OREN AND UZY SMILANSKY, Proof. i. Assume that i is an interior vertex in g. Hene, the signs of f j for all j i are the same as the sign of f i. This is not ompatible with (7) f j = (λ v)f i j i for λ > v. Hene g annot have any interior verties. ii. Assume that the subgraph g onsists of a single vertex i. Thus on all its neighbors j i, the sign of f j is different from the sign of f i. This is not ompatible with (8) f j = (v λ)f i j i for λ < v. Hene g annot onsist of a single vertex. iii. Denote the omplement of the nodal domain (subgraph) g in V by g. For all the verties i in g (9) (Lf) i = vf i C j,i f j C l,i f l = λf i. j g l g Summing over i g we get: (10) (v λ) f i = C j,i f j + C l,i f l i g i g j g l g Assuming for onveniene that f i are positive for i g, the rightmost sum in the equation above is non positive, and therefore (11) (v λ) f i C j,i f j = v i f i ˆv f i. i g i g j g i g i g Here, v i = j g C j,i is the valeny (degree) of the i th vertex in g, and ˆv denotes the largest valeny in the subgraph. Sine it is assumed that λ < v k we get (1) k < ˆv, whih ompletes the proof. The ase λ = v deserves speial attention. As long as the nodal domain under study has no vanishing entries, it annot onsist of a single vertex nor an it have interior verties. Namely, for λ = v, statements i and ii of Theorem 3.1. are valid simultaneously. Otherwise, one should treat separately the strong and the weak ounts. For the strong ount, and λ = v a nodal domain annot have an interior vertex. However, using the weak ount for λ = v one finds that no single vertex domains an exists, as in Theorem 3.1.ii. Item iii of Theorem 3.1 an be used to provide a λ dependent bound on the number of nodal domains of eigenvetors orresponding to eigenvalues λ < v. Define the integer k as k = v λ. Theorem 3.1.iii implies that every nodal domain oupies at least k + verties. Thus, their number is bounded by V k+. Courant theorem guarantees that the number of nodal domains is bounded by the spetral
7 NODAL DOMAINS ON GRAPHS  HOW TO COUNT THEM AND WHY? 7 ount N(λ). This information together with the known expression for the expetation value of N(λ) over the ensemble of random graphs, enable us to show that for large v and V, the bound V k+ is more restritive than the Courant bound. Unlike Pleijel s result, this bound is not uniform for the entire spetrum, and it applies only to the lower half of the eigenvalues with λ < v. Theorem 3.1 an be easily extended to the nodal properties of the eigenvetors of the generalized Laplaian, provided that the weights at eah vertex sum up to a onstant v whih is the same for the entire graph. 4. How to ount nodal domains on graphs? When disussing nodal domains ounting, we must make a lear distintion between algorithmi and analyti methods. In the first lass, we inlude omputer algorithms. They vary in effiieny and reliability, but they have one feature in ommon, namely, that the number of nodal domains is provided not as a result of a omputation, but rather, it follows from a systemati ounting proess. The most widely used method is the HoshenKopelman algorithm (HK) for ounting nodal domains on dimensional domains [30]. Analyti methods provide the number of nodal domains as a funtional of the funtion and the domain under study. The funtional might be quite ompliated, and not effiient when implemented numerially. An example of an analytial method for nodal domains ounting in one dimension, is given by (13) ν = b a δ (f(x)) df(x) dx dx + 1, where the nodal domains of f(x), in the interval [a,b] are provided (assuming that f(a)f(b) 0). While ounting in 1d is simple, there is no analyti ounting method for omputing the number of nodal domains in higher dimensions: the ompliated onnetivity allowed in high dimensions renders the ounting operation too non loal. Graphs, whih are in some sense intermediate between one and two dimensions still allow several analyti ounting methods whih we disuss here. An example of an analytial ount is given by (6). The HK algorithm is well suited for graphs whih are grids. However, it is not as effiient when the graph under study is highly irregular. Although the HK algorithm fails for very omplex graphs, other algorithms, alled labeling algorithms, display linear effiieny ([46],[47]). Method III. in the following list, in addition of providing an analytial expression for the nodal domain ount, an also be implemented as a omputer algorithm. We show that it performs as effiiently as the labeling algorithm. The ounting methods that we present here are aimed for the disrete ounting of both disrete and metri graphs. In what follows, we assume that a vetor f is assoiated to the vertex set with entries f i. The nodal domains are defined with respet to f Method I.  Counting nodal domains in terms of flips. We define a flip as a bond on the graph whih onnets verties of opposite signs with respet to a vetor f. The sign vetor of f, denoted by f, is defined by f i Sign(f i ). For the time being, it is assumed that f has no zero entries. The general situation will
8 8 RAM BAND, IDAN OREN AND UZY SMILANSKY, be disussed later. We denote the set of flips on the graph by F(f): (14) F(f) = {(u,v) B f v f u < 0}. The ardinality of F(f) will be denoted by F(f). Lemma 4.1. The number of flips of a sign vetor f, an be expressed as (15) F(f) = F( f) = 1 4 ( f,l f). Proof. Using: (16) (17) ( f,l f) = 1 ( f v f u ) v u ( f v f u ) 4, if f v and f u have opposite signs = 0, if f v and f u have the same sign Using the number of flips, one an get an expression for the number of nodal domains: Theorem 4.. Given a onneted graph G on V verties, B bonds (and r yles) and a vetor f, then: (18) ν(f) = 1 4 ( f,l f) + V B + l(f) = 1 4 ( f,l f) (r l(f)) + 1 where l(f) is the number of independent yles in G of onstant sign (with respet to f). The seond equality above is based on equation (1). Proof. Let us remove all the flips from the graph. We are now left with a possibly disonneted graph G. There is a bijetive mapping between omponents of G and nodal domains of G. Hene, the number of omponents in G is equal to the nodal domains ount of G with respet to f. Let the number of nodal domains in G be denoted by ν(f). Using (1), it is lear that for the i th omponent (where i = 1,,...,ν(f)): r i = B i V i + 1 where r i, B i and V i are the number of yles, bonds and verties of the i th omponent, respetively. It is also lear that all the yles in G are of onstant sign, sine there are no flips in G. Thus, by our notation r i = l i. Let us sum over the omponents: ν ν (19) l(f) = l i = (B i V i + 1) = (B F(f)) V + ν(f) Combining (15) with (19), we get (18).
9 NODAL DOMAINS ON GRAPHS  HOW TO COUNT THEM AND WHY? 9 (18) is valid only for vetors f with no zero entries. In order to be able to handle a zero entry in f, we must perform a transformation on the graph. For a strong nodal ount, we simply delete all the zero verties along with the bonds onneted to them from the graph, and then apply (18) on the new graph (with the new Laplaian). For a weak nodal ount we replae eah zero vertex by two verties, one positive and one negative (not onneted to eah other), and onnet them to all verties whih were onneted to the original one. Now we an apply (18), and get the desired result. Notie that this orretion fails in the ase of a zero vertex whose neighbors are of the same sign (in this ase an artifiial nodal domain is added). However, the situation above an not our for an eigenvetor of a disrete Laplaian. Using (18), we an write some immediate onsequenes: (0) (1) F(f n ) + 1 r ν n F(f n ) + 1 n r 1 F(f n ) n + r 1 (0) results from the obvious fat that 0 l r, while (1) is a onsequene of Courant s nodal domains theorem and Berkolaiko s theorem whih states that n r ν n. In order to make use of (18), one must ompute l(f) whih is not given expliitly in terms of f. Thus, it annot be onsidered as an analyti ounting method, nor does it offer omputational advantage (There is no known effiient algorithm whih ounts all the yles of onstant sign with respet to f). However, it offers a useful analytial tool for deriving other results, and it makes a useful onnetion between various quantities defined on the graph. 4.. Method II. Partition funtion approah. Foltin [3] derived a partition funtion approah to ounting nodal domains of real funtions in two dimensions. It an be adapted for graphs in the following way: Eah vertex, i, is assigned an auxiliary spin variable s i where s i = ±1 (a so alled Isingspin). Thus, given a ertain funtion f on the graph, eah vertex is assigned with two spins s i and f i. Let s denote the auxiliary spin vetor: s = (s 1,s,...,s V ). Foltin introdued a weight to eah onfiguration of the spins model. It assigns the value 1 to onfigurations in whih all spins s i belonging to the same nodal domain (with respet to f) are parallel, while spins of different domains might have different values. The weight reads: () w(f,s) = i,j C i,j [1 1 + f i fj 1 s is j ]. It an be easily heked that this form satisfies the requirements stated above: The weight an take the values one or zero. It is one if and only if eah fator in the produt is equal to one. A ertain fator is one, in either one of the two ases: if f i f j (i,j belong to different domains)  this allows the Isingspins in different domains to be independent of eah other. The seond ase is if i,j are in the same domain, fi = f j and the orresponding Isingspins are equal, s i = s j. Let us now sum over all possible spin onfigurations {s i } to get the partition funtion (3) Z(f) {s} w(f,s).
10 10 RAM BAND, IDAN OREN AND UZY SMILANSKY, For the onfigurations whose weight has the value one, the Isingspins an take both signs, as long as they are all equal on a domain. Different domains are independent of eah other. Hene, the total number of suh onfigurations is: (4) Z(f) = ν(f), where ν(f) is the number of nodal domains of the vetor f. The nodal domains ount is: (5) ν(f) = 1 lnz(f) 1.44ln Z(f). ln The partition funtion approah provides an expliit formula for the number of nodal domains, and therefore it belongs to the analyti and not to the algorithmi ounting methods. As a matter of fat, it is highly ineffiient for pratial omputations. It involves running over all possible spin onfigurations {s i }, where s i = ±1. There are V different onfigurations, and as V inreases the effiieny deteriorates rapidly. The partition funtion approah an be used as a basis for the derivation of some identities involving the graph and a vetor f defined on it. It is onvenient to introdue the following notations: b (i,j) ϕ b 1 + f i fj σ b 1 s is j Where f and s are as before, and b is a direted bond (from i to j). We generalize the partition funtion by introduing a new parameter x into the definitions (6) (7) (8) w(f,s;x) = b B[1 ϕ b σ b x] Z(f;x) {s} Z(f;1) = ν(f) w(f,s;x) = {s} (1 ϕ b σ b x) where x an assume any real or omplex value. At x = 1, the generalized partition funtion is idential to (3). Let us now perform the summation over all the vetors s, and ompute the oeffiient of x k. To get all the ontributing terms we have to sum over all hoies of k brakets from (7), in whih x appears. Non vanishing ontributions our whenever both ϕ b and σ b are equal to one. Sine we are only summing over s, we only need to hek when σ b = 1. This happens if and only if the s vetor has a flip on the bond b. Sine we hoose k brakets (whih is equivalent to hoosing k bonds), we need to ount how many s vetors have flips on all these k bonds. The signs of those s vetors on bonds whih are not ontained in this hoie of k bonds are irrelevant. If we observe the hoie of k bonds (b 1,...,b k ), we notie that eah onneted omponent, within this hoie, ontributes a fator of, sine the symmetry of turning eah plus to minus and vie versa, does not hange the flip properties. Using (1) we see that the number of onneted omponents with respet to the hoie of k bonds is: Co(b 1,...,b k ) = V k + r(b 1,...,b k ), where r(b 1,...,b k ) is the number of independent yles that are ontained in this hoie. b B
11 NODAL DOMAINS ON GRAPHS  HOW TO COUNT THEM AND WHY? 11 Finally we notie that a yle of odd length annot have a flip on all its bonds, so we will not sum over hoies of b 1,...,b k whih ontain a yle of odd length. Thus, the sum over s yields: B k Z(f; x) = (ϕ bi ) V k+r(b1,...,bk) ( 1) k x k (9) k=0 b 1,...,b k B = B V ( 1) k k=0 k b 1,...,b k B k (ϕ bi ) r(b1,...,b k) x k Where stands for summation on all the possibilities to hoose k bonds b 1,...,b k B b 1,...,b k B suh that the subgraph they form do not ontain an odd yle. We an now derive some immediate properties of the generalized partition funtion. To start, we ompute the leading four derivatives at x = 0 to demonstrate the ounting tehniques involved. Some more effort is required to ompute the higher derivatives. (30) (31) (3) (33) (34) Z(f;0) = V Z (1) (f;0) = V 1 ϕ b = V 1 (B F(f)) b B Z () (f;0) =! V b 1,b B Z (3) (f;0) = 3! V 3 Z (4) (f;0) = 4! V 4 ( ) B F(f) (ϕ bi ) = V 1 b 1,...,b 3 B b 1,...,b 4 B 3 [( ) ] B F(f) (ϕ bi ) = 3! V 3 C (ϕ bi ) [( ) ]} B F(f) = 4! {C V C 4 4 C 3 [( ) ] B F(f) = 4! V 4 + C 4 4 C 3 Where C 3 is the number of triangles of onstant sign, C 4 is the number of yles of length 4 of onstant sign, and C 3 = C 3 (B F(f) 3) is the number of hoies of 4 nonflips bonds whih ontain a triangle. In the evaluation of the funtion and its first three derivatives we have used the identity n 3 r(b 1,...,b n ) = 0 when b 1,...,b n ontain no odd yles. The partition funtion is a polynomial of degree less than or equal to the number of bonds. For a graph G with B bonds and P onneted omponents, the polynomial is of degree B if and only if G is bipartite and the funtion f is of onstant sign on eah onneted omponent. This follows from two observations: first, a well known theorem in graph theory states that a graph G is bipartite if and only if it ontains no yles of odd length. Thus we an hoose all the bonds in G without having an odd yle ontained in this hoie. Seond, unless f is of onstant sign on eah onneted omponent, we will enounter a flip and hene the multipliation over all ϕ bi will vanish. If G is bipartite, the oeffiient of x B is P.
12 1 RAM BAND, IDAN OREN AND UZY SMILANSKY, While the value of the polynomial at x = 1 has an immediate appliation through (8), we an evaluate it for other values. Let us hoose f to be a vetor of onstant sign. This way ϕ bi = 1 for all i = 1,,...,B. Hene (9) redues to: (35) B V ( 1) k x k. k=0 k b 1,...,b k B r(b1,...,b k) On the other hand, (7) equals: (36) (1 σ b x). {s} b B Choosing x = n + 1 where n Z and using the fat that (35) and (36) are equal, we get: (37) 1 B V n F(s) ( 1) k = k r(b1,...,b k) ( n + 1) k. {s} k=0 b 1,...,b k B For random vetors s, uniformly distributed, the left hand side an be interpreted as the average over this ensemble of the quantity: n F(s). Let us hoose two speial values for n. If we hoose n = 1, the left hand side is just the differene between the probability that a random vetor s on the graph has an even number of flips, and an odd number of flips: prob(f(s) =even)prob(f(s) =odd). The right hand side is: (38) B ( 1) k r(b1,...,bk). k=0 b 1,...,b k B For n =, we get: (39) 1 V F(s) = {s} B k=0 1 k r(b1,...,bk). b 1,...,b k B An example of the use of the polynomial ould be in proving that on a tree, the probability of an even number of flips is equal to the probability of an odd number of flips. We use Eq. (38) and see that this differene of probabilities is equal to: B k=0 ( 1)k( B k) = 0. Other possible identities we an derive are for the omplete graph, K V. A funtion f of onstant sign indues one nodal domain on K V, while any other funtion indues two nodal domains. Therefore, for x = 1: (40) (41) V V ± 1 f {±1} V V B k=0 B ( 1) k k=0 k b 1,...,b k B ( 1) k k b 1,...,b k B ( k ) ϕ bi (f) r(b1,...,b k) = r(b1,...,b k) = 4
13 NODAL DOMAINS ON GRAPHS  HOW TO COUNT THEM AND WHY? 13 Where ϕ bi (f) = 1+fufv for b i = (u,v). Equivalently, running over all the funtions f (inluding those of onstant sign) we get: B V ( 1) k ( k ) (4) ϕ bi (f) r(b1,...,b k) = 4( V 1) f {±1} V k=0 k b 1,...,b k B 4.3. Method III. Breaking up the graph. We begin again by deleting all the flips from the graph G. This way we are left with a (possibly) disonneted graph, G in whih eah onneted omponent orresponds uniquely to a nodal domain in the original graph. The onnetivity matrix C and the disrete Laplaian L of G are given by (43) (44) C ij = C ij 1 + f i fj L ij = C V ij + δ ij Where f is the sign vetor, and it is assumed for the moment that none of the entries of f vanish. We now make use of the theorem whih states that the lowest eigenvalue of the Laplaian is 0 with multipliity whih equals the number of onneted omponents in the graph. Therefore, finding the nodal domains ount redues to finding the multipliity of zero as an eigenvalue of L. An analyti ounting formula an be derived by onstruting the harateristi polynomial of L: (45) det(λi V L) The multipliity of its 0 eigenvalue provides the nodal domains ount: (46) ν(f) = lim λ 0 λ d dλ lndet(λi V L). The appliation of the method an be extended to ases where some entries of f vanish. Then we would like to ount both weak and strong nodal domains. For the strong nodal ount, we an redue the dimension of our problem. We simply remove the zero entries from f, along with the orresponding rows and olumns in L. Now we onstrut L, and ontinue as before, with the new matrix L of dimension (V z) (V z), z being the number of zero entries in f. The weak nodal domains ount is obtained in the opposite manner: sine a zero entry partiipates both in positive and negative nodal domains, we replae eah zero vertex by two verties, one positive and one negative, both onneted to all verties adjaent to the original one. Thus, for the weak nodal ount, the dimension of the new problem is (V + z) (V + z), z is defined as before. Noting again that this fails in the ase of a zero vertex whose neighbors are of the same sign (see orresponding remark in subsetion 4.1). This method of ounting, whih provides the analytial expression (46) for the nodal ount, is also the basis for a omputational algorithm whih turns out to be very effiient. It relies on the effiieny of state of the art algorithms to ompute the spetrum (inluding multipliity) of sparse, real and symmetri matries. To k=1 C ik
14 14 RAM BAND, IDAN OREN AND UZY SMILANSKY, estimate the dependene of the effiieny on the dimension V of the graph, we have to onsider the osts of the various steps in the omputation. The onstrution of the matrix L, takes O(V ) operations, and storing the information requires O(V ) memory ells as well. It takes O(V α ) operations to find all its eigenvalues where α.3 (and at worst ase α 3) [9]. In figure 1, this polynomial dependene is shown for graphs of two different onnetivity densities, with r V equals 0.5 and 5. In this figure the logarithm of the time needed to find all eigenvalues of L (defined for a random vetor), is plotted against the logarithm of the number of verties. The slope whih is the exponent of the polynomial dependene is smaller than 3. The eigenvalues in these two examples were attained using the Matlab ommand eig. As will be shown below, there are more effiient ways of finding the spetrum of sparse, real and symmetri matries. Thus, the effiieny stated above an be improved for graphs with sparse Laplaians. (a) (b) Figure 1. The time it takes to ompute the spetrum of L as a funtion of the number of verties, for two different onnetivity densities: (a) r V = 0.5 (b) r V = 5. Finally, we would like to show that the present method an be applied for ounting nodal domains of funtions defined on two dimensional grids and that its effiieny is omparable to that of the ommonly used HK algorithm. Given a funtion f on a two dimensional domain, we have to ompute its values on a retangular grid with V V points. The HK algorithm ounts the nodal domains in O(V ) operations [30]. Using our method, we onsider the retangular grid as a graph with V verties. Assuming for simpliity periodi boundary onditions, the valeny of all the verties is 4. The orresponding L matrix is a V V matrix whih is sparse (as long as V 4), and due to the periodi boundary onditions it takes the expliit form: (47) L i,j = 4δ i,j δ i,j 1 δ i,j+1 δ i V,j δ i+v,j. Thus, storing L takes only O(V ) memory ells, and onstruting L takes O(V ) operations. We mentioned above that for a general real symmetri matrix, the number of operations needed is O(V α ), where α.3 and at worst ase α 3. However, the sparse nature of L signifiantly simplifies the problem. The most
15 NODAL DOMAINS ON GRAPHS  HOW TO COUNT THEM AND WHY? 15 wellknown eigenvalue method for sparserealsymmetri matries is the Lanzos method. In addition, in reent years, new effiient algorithms were disovered for the same problem. In [31], it is proven that finding the eigenvalues of a sparse symmetri matrix takes only O(V ) operations. Combining the osts, we find that it takes our algorithm O(V ) operations in order to ompute the nodal domains ount, and therefore it is omparable in effiieny to the HK algorithm. As mentioned earlier, the labeling algorithms also display linear effiieny ([46],[47]). The labeling algorithms have the advantage that they are simpler in a sense, and that they are implemented quite easily as omputer programs. In addition, the labeling algorithms maintain their linear effiieny even for graphs with dense Laplaians. It is worth mentioning, however, that our algorithm has the advantage that it provides an analyti expression of the nodal domains ount (46) Method IV. A geometri point of view. The ounting method proposed here uses a geometri point of view whih starts by onsidering the V dimensional Eulidean spae, and dividing it into V setors using the following onstrution. Consider the V vetors e (α) = (e (α) 1,e (α),...,e (α) V ) where e(α) i = {1, 1}, α = 1,,..., V. A vetor x R n is in the setor α if x e (α) = n x i. In two dimensions, the setors are the standard quadrants. We shall refer to the vetors e (α) as the indiators. Given a graph G with V verties, we partition the V indiators into disjoint sets: γ n = {e (α) : ν(e (α) ) = n} where ν(e (α) ) denotes the nodal domains ount of the indiator e (α) with respet to G. As shown before {γ n } V χ +, where χ is the hromati number of G, and also some of the γ is might be empty. Let f be a vetor with nonzero entries defined on the vertex set of G. Then, the main observation is that ν(f) = n if and only if: (48) V ˆδ(< e α f > f i ) = 1, e α γ n where, ǫ (49) ˆδ(x) = lim ǫ 0 x sin x { 1, if x = 0 ǫ = 0, if x 0. In other words, by finding the setor to whih f belongs and knowing from a preliminary omputation the number of nodal domains in eah setor, one obtains the desired nodal ount. Thus, the present method requires a preliminary omputation in whih the setors are partitioned into equinodal sets γ n. This should be arried out one for any graph. Therefore the method is useful when the nodal ounts of many vetors is required. In several appliations, one is given a vetor field (of unit norm for simpliity) f S V 1 whih is distributed on the V 1sphere with a given probability distribution p(f), and one is asked to ompute the distribution of nodal ounts, (50) P(n) = p(f)ˆδ(ν(f) n)d V 1 f. S V 1
16 16 RAM BAND, IDAN OREN AND UZY SMILANSKY, In suh ases, the preliminary task of omputing the equinodal pays off, and one obtains the following analyti expression for the distribution of the nodal ounts. (51) P(n) = p(f)d V 1 f V ˆδ(< e α f > f i ) = S V 1 e α γ n p(f)δ(1 f )d V f V ˆδ(< e α f > f i ) R V e α γ n where: p(f) = p( f f ) f. (51) an also be formulated as: (5) V P(n) = p(f β )δ(1 f )d V f V ˆδ(< e α f β > f i ) = β=1 f i 0 e α γ n + p(f β )δ(1 f )d V f V ˆδ(< e α f β > f i ) e α γ n e β γ n Where + = f i 0 means integration on the first setor (the vetors with all entries positive) And f µ is the dot produt of f and e µ : f µ = (f 1 e µ 1,f e µ,...,f V e µ V ) Equations (51) and (5) are the general equations governing the nodal domains ount distribution. In order to make further progress, we need to speify the distribution from whih f is taken. This means that we need to speify p(f) in (51) for example. Let us disuss two examples: A uniform distribution over the V 1 dimensional sphere: In this ase, we an solve Equation (51) and get that P(n) = γn. Note that for a tree, we an solve V this problem by other means. Using (18), we see that for a tree, the number of nodal domains is equal to the number of flips plus one. Sine f is taken from the uniform distribution, then the probability of a flip is half. The number of flips in a vetor f is thus a binomial variable: F(f) Binomial(N,p) with N = V 1 is the number of bonds, and p = 1. For large enough V this approahes the Gaussian distribution: F(f) Gaussian(µ,σ ) with µ = V 1 and σ = V 1 4. From this result we an infer that: (53) (54) V +1 (n P(n) exp ( ) ) π(v 1) V 1 V +1 V +1 (n γ n exp ( ) ) π(v 1) V 1 For the other extreme, the omplete graph, K V, the only possible nodal domains ounts are one and two [8]. The vetors whih yield a nodal domains ount of one are vetors of onstant sign. All other vetors yield a nodal domains ount of two. Indeed, using (51) or (5) it is easy to be onvined that for the omplete graph, γ 1 = while γ = V. All other γ s are empty. Miroanonial ensemble: In this ase the vetors f are uniformly distributed on
17 NODAL DOMAINS ON GRAPHS  HOW TO COUNT THEM AND WHY? 17 the energy shell, where we an also define a measurement tolerane fator, : (55) p E (f) = δ(e < f L f > )δ(1 f ) d S V 1 fδ(e < f L f > ) V 1 In order to make use of this ensemble, further work must be done, for example, a natural way to order the funtions of the ensemble. 5. The resolution of isospetrality There are several known methods to onstrut isospetral yet different graphs. A review of this problem an be found in ([33]). The onditions under whih the spetral inversion of quantum graphs is unique were studied previously. In [38, 39] it was shown that in general, the spetrum does not determine uniquely the length of the bonds and their onnetivity. However, it was shown in [35] that quantum graphs whose bond lengths are rationally independent an be heard  that is  their spetra determine uniquely their onnetivity matries and their bond lengths. This fat follows from the existene of an exat trae formula for quantum graphs [40, 41]. Thus, isospetral pairs of non ongruent graphs, must have rationally dependent bond lengths. An example of a pair of metrially distint graphs whih share the same spetrum was already disussed in [35]. The main method of onstrution of isospetral pairs is due to Sunada [34]. This method enabled the onstrution of the first pair of planar isospetral domains in R [36] whih gave a negative answer to Ka s original question: Can one hear the shape of a drum? [37]. Later, it was shown that all the known isospetral domains in R [4, 43] whih were also onstruted using the Sunada method have orresponding isospetral pairs of quantum graphs [44]. An example of this orrespondene is shown in figure. As mentioned in the introdution, it is onjetured [7, 19] that nodal domains sequenes resolve between isospetral domains. For flat tori in 4d, this was proven [8]. We present here three additional known results for the validity of the onjeture for graphs. The first result is for the quantum graphs shown in figure (). Both graphs of this isospetral pair are tree graphs and therefore have the same metri nodal ount µ n = n ([15]). This demonstrates the need to use the disrete nodal ount in order to resolve isospetrality in this ase. Indeed numerial examination of this ase shows that for the first 6600 eigenfuntions there is a different disrete nodal ount for 19 % of the eigenfuntions. Similar numerial results exist for two other pairs of isospetral graphs that are onstruted from the isospetral domains in [4, 43]. The exat results are desribed in [19]. Another result is also in the field of quantum graphs [19]. The graphs in figure 3 are the simplest isospetral pair of quantum graphs known so far. The simpliity of these graphs enables the omparison between the nodal ounts of both graphs. It was proved that the nodal ount is different between these graphs for half of the spetrum. This result was proved separately for the disrete ount and for the metri ount. The proof does not ontain an expliit formula for the nodal ount but rather deals with the differene of the nodal ount between the graphs averaged over the whole spetrum.
18 18 RAM BAND, IDAN OREN AND UZY SMILANSKY, I II (a) a b (b) a b b a a b a b b a a b a b II b a b a a b b a b a b a a b I () Figure. (a) Planar isospetral domains of the 7 3 type. (b) Reduing the building blok to a 3star. () The resulting isospetral quantum graphs. I II Figure 3. The isospetral pair with boundary onditions. D stands for Dirihlet and N for Neumann. The bonds lengths are determined by the parameters a,b, Examining the nodal sequenes for the graph II for various values of the length parameters a,b,, we observed that the formula (56) µ II n = n 1 1 ( 1) b+ a+b+ n reprodues the entire data set without any flaw 1. Assuming it is orret (whih is not yet proved rigorously) we first see that it provides an easy explanation for the previously disussed result that for rationally independent values for the parameters a,b, one gets that µ II n n for half of the spetrum. This, together with µ I n = n (sine graph I is a tree) we see again that for half of the spetrum the nodal 1 This result was obtained with A. Aronovith.
19 NODAL DOMAINS ON GRAPHS  HOW TO COUNT THEM AND WHY? 19 domain sequenes are different. Expression (56) is a periodi funtion of n with period proportional to the length of the only loop orbit on the graph (the length is measured in units of the its total length). It an be expanded and brought to a form whih is similar in struture to a trae formula where the length of this orbit and its repetitions are the osillation frequenies. A similar trae formula for the nodal ounts of the Laplaian eigenfuntions on surfaes of revolution was reently derived [6]. Finally, we diret our attention to disrete Laplaians. It was reently shown [8] that if G and H are two isospetral graphs where one of them is bipartite and the other one is not (without loss of generality, G is bipartite and H is not). Then for the eigenvetor orresponding to the largest eigenvalue λ V, their nodal domains ount will differ (ν(g) = V, ν(h) < V ). The proof of this theorem is based on another interesting result derived in [8]: Denote by f V the eigenvetor orresponding to the largest eigenvalue of the Laplaian of a onneted graph G. Then ν(f V ) = ν G = V, if and only if G is bipartite. Figure 4. illustrates this result. Figure 4. The upper figure presents a pair of isospetral graphs taken from [6]. Graph G, on the right is bipartite, whereas graph H, on the left, is not. The lower figure presents the nodal domains ount, ν(f n ) vs. the index n.
20 0 RAM BAND, IDAN OREN AND UZY SMILANSKY, 6. Summary and open questions In spite of the progress ahieved reently in the study of nodal domains on graphs, there are several outstanding open problems whih all for further study. We list here a few examples. Of fundamental importane is to find out whether there exists a trae formula for the nodal ount sequene of graphs, similar to the one derived in [6] for surfaes of revolution. The losest we reahed this goal is for the graph II in the previous setion, where (56) ould be expanded in a Fourier series. However (56) was dedued numerially but not proved. One a nodal trae formula is available, it ould be ompared to the spetral trae formula [41] and might show the way to prove or negate the onjeture ([7, 19]) that ounting nodal domains resolves isospetrality. The onjeture mentioned above an be addressed from a different angle. One may study the various systemati ways to onstrut isospetral pairs and investigate the relations between the onstrution method and the nodal ount sequene of the resulting graphs. Suh an approah worked suessfully for the dihedral graphs presented in the previous setion [19]. Another open question whih naturally arises in the present ontext: Can one find graphs whose Laplaians have different spetra but the nodal ount sequenes are the same? A positive answer is provided for tree graphs ([15]). Are there other less trivial examples of isonodal yet not isospetral domains? It follows from Berkolaiko s theorem [7] that the number of nodal domains (both metri and disrete) of the n th eigenfuntion is bounded in the interval [n r,n]. We an thus investigate the probability to have a nodal ount ν n = n r (for 0 r r). This probability, whih is defined with respet to a given ensemble of graphs, is denoted by P( r). It is defined for disrete graph laplaians as: (57) P( r) 1 V # { 1 n V : ν n = n r}. The orresponding quantity for metri Laplaians is: (58) N(K) # {n : k n K} P( r;k) 1 N(K) # { n N(K) : µ n = n r} P( r) lim K Here, stands for the expetation with respet to the ensemble. New questions arise from the investigation of the relation between the onnetivity of the graph and the nodal distribution P( r). Can one use the information stored in P( r) to gain information on the graphs e.g., the mean and the variane of the valeny (degree) distribution of the verties in the graphs? Many of the results we have presented, have analogues in Riemannian manifolds (whih in most ases, were disussed earlier)  for example, Courant s theorem was originally formulated for manifolds. One an searh for other analogues, and a good example is the CourantHerrmann Conjeture (CHC). For manifolds the CHC onjeture states that any linear ombination of the first n eigenfuntions divide the domain, by means of its nodes, into no more than n nodal domains. Gladwell and Zhu [45] have shown that in general there is no disrete ounterpart to the CHC. However, we an still ask for whih lasses of graphs does the CHC still hold?
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