Nodal domain theorems for $p$-Laplacians on signed graphs

We establish various nodal domain theorems for $p$-Laplacians on signed graphs, which unify most of the existing results on nodal domains of graph $p$-Laplacians and arbitrary symmetric matrices. Based on our nodal domain estimates, we obtain a higher order Cheeger inequality that relates the variational eigenvalues of $p$-Laplacians and Atay-Liu's multi-way Cheeger constants on signed graphs. In the particular case of $p=1$, this leads to several identities relating variational eigenvalues and multi-way Cheeger constants. Intriguingly, our approach also leads to new results on usual graphs, including a weak version of Sturm's oscillation theorem for graph $1$-Laplacians and nonexistence of eigenvalues between the largest and second largest variational eigenvalues of $p$-Laplacians with $p>1$ on connected bipartite graphs.


Introduction
Graph p-Laplacian is a natural discretization of the continuous p-Laplacian on Euclidean domains, and it is also a simple nonlinearization of the Laplacian matrix.The spectrum of the graph p-Laplacian is closely related to many combinatorial properties of the graph itself; and its eigenpairs, reveal important information about the topology and geometry of the graph.For example, similar to the original Euclidean p-Laplacian and graph linear Laplacian, the p-Laplacian on graphs has some important relations to Cheeger cut problem and shortest path problem on graphs.Just as the Laplacian matrix which has been successfully used in diverse areas, the graph p-Laplacian has been also widely used in various applications, including spectral clustering [10,33,55,56], data and image processing problems, semi-supervised learning and unsupervised learning [50,55,56].Much recent work has shown that algorithms based on the graph p-Laplacian perform better than classical algorithms based on the linear Laplacian in solving these practical problems in image science.
The theoretical aspects of p-Laplacians on graphs and networks are still not well understood due to the nonlinearity.Among several progresses in this direction, a remarkable development is that the second eigenvalue has a mountain-pass characterization and it is a variational eigenvalue which satisfies the Cheeger inequality [3,10].Another important result is the nodal domain count for graph p-Laplacians, including an interesting relation that connects the nodal domains of the p-Laplacian and the multi-way Cheeger constants on graphs [49].For the limiting case p = 1, the spectral theory for graph 1-Laplacian was proposed by Hein and Bühler [33] for 1-spectral clustering, and was latter studied by Chang [13] from a variational point of view.For example, Cheeger's constant, which has only some upper and lower bounds given by the second eigenvalues of p-Laplacians with p > 1, equals the second eigenvalue of graph 1-Laplacian [13,33].Moreover, any Cheeger set can be identified with any strong nodal domain of any eigenfunction corresponding to the second eigenvalue of graph 1-Laplacian.
To some extent, nodal domain theory provides a good perspective for understanding the spectrum of graph p-Laplacians.Indeed, various versions of discrete nodal domain theory have been developed in different contexts.A very useful context should be the signed graphs, whose spectral theory has led to a number of breakthroughs in theoretical computer science and combinatorial geometry, including the solutions to the sensitivity conjecture [35] and the open problems on equiangular lines [11,36,37].In addition, signed graphs have many other practical applications on modeling biological networks, social situations, ferromagnetism, and general signed networks [5,6,32].Therefore, it should be natural and useful to develop a general spectral theory that includes nodal domain theorems on signed graphs.Along this line, Ge and Liu [31] provided a definition of the strong and weak nodal domains on signed graphs, which is compatible with the classical one in [23] on graphs.They also obtained sharp estimates of the number of strong and weak nodal domains for generalized linear Laplacian on signed graphs.We notice that estimates of strong nodal domains on signed graphs has been established in an earlier work of Mohammadian [43], see [31,Remark 3.12].For more details and historical background of nodal domain theory, we refer the readers to [31].We particularly mention that the results in Fiedler's classical 1975 paper [27] can be considered as nodal domain theorems on signed trees (see [31,Section 5]).In 2013, Berkolaiko [8] and Colin de Verdière [18] computed the nodal count of edges on signed graphs by allowing the signs of each edge to become complex.See Remark 2.2 for more detailed comments.
The combination of signed versions and nonlinear analogs of nodal domain theorems is the main focus of this paper.To the best of our knowledge, the p-Laplacian on signed graphs has not been well studied.A related research was given in [38] for p-Laplacians on oriented hypergraphs, which includes the p-Laplacian on signed graphs as a special case.However, that paper does not focus on the nodal domain property, so there are no sufficiently in-depth results on nodal domain theorems for p-Laplacians on signed graphs.
In this paper, we systematically establish a nodal domain theory for p-Laplacians on signed graphs, which unifies the ideas and approaches from these recent works [24,31,38,49].Based on our nodal domain estimates, we also obtain a higher order Cheeger inequality that relates the variational eigenvalues of p-Laplacians and Atay-Liu's multi-way Cheeger constants on signed graphs [4].Although these results appear to be formally similar to that in [24,49], there are several key differences in both results and approaches.First, our upper bounds for the number of dual nodal domains for p-Laplacians on signed graphs are new, and the proof relies heavily on the intersection property of Krasnoselskii genus.In particular, for p > 1, the estimate of the number of dual weak nodal domains, and the bound on the number of dual strong nodal domains of the k-th eigenfunction with minimal support, further require the odd homeomorphism deformation lemma in Struwe's book [48]; while the case of p = 1 should be treated separately by using the localization property.It is worth noting that a cautious analysis gives us a stronger result for the signed 1-Laplacian case, which is also new for graph 1-Laplacian.Second, the approach we use to obtain the lower bound estimates for the number of strong nodal domains, further relies on a duality argument by considering the quantity S(f )+S(f ), which is similar to the linear case in [31], but the nonlinear estimate requires more subtle techniques.Third, the k-way Cheeger inequality connecting variational eigenvalues of p-Laplacians and Atay-Liu's k-way Cheeger constants on signed graphs is essentially new, although the proof is not difficult to anyone who is familiar with analysis or spectral graph theory.Interestingly, this result also reveals that variational eigenvalues of the 1-Laplacian on signed graphs are very closely related to certain combinatorial quantities on signed graphs.Fourth, it should be noted that many of the nodal domain properties of p-Laplacians are different on graphs of different signatures.For example, on a balanced graph, the second eigenfunction has exactly two weak nodal domains (see [24]), which is not always the case on an unbalanced graph, see Example 3.1.Very interestingly, we prove a nonlinear Perron-Frobenius theorem for p-Laplacians on antibalanced graphs, that is, the eigenfunction corresponding to the largest eigenvalue is positive everywhere or negative everywhere.Moreover, the eigenfunction corresponding to the largest eigenvalue is unique up to a constant multiplication.However, this does not hold for p-Laplacians on balanced graphs.
Even on the usual graphs, our theorems directly derive at least two new results: • Any eigenfunction corresponding to the k-th variational eigenvalue λ k (such that λ k > λ k−1 ) of the graph 1-Laplacian with minimal support has at least k + r − 2 zeros, where r is the variational multiplicity of λ k (see Theorem 4).Recall Sturm's oscillation theorem, which says that the k-th eigenfunction of the second-order linear ODE has exactly (k − 1) zeros.Our result actually shows that the k-th variational eigenfunction of the graph 1-Laplacian with minimal support has at least (k − 1) zeros.Therefore, in a sense, we are actually building a weak version of Sturm's theorem for the graph 1-Laplacian.
• When p > 1, there are no other eigenvalues between the largest and the second largest variational eigenvalues of the graph p-Laplacian on connected bipartite graphs (see Corollary 5.1).This new phenomenon can be seen as a dual version of the classic result that there are no other eigenvalues between the smallest and the second smallest variational eigenvalues of the graph p-Laplacian.
The paper is structured as follows.In Section 2, we collect preliminaries on p-Laplacians and signed graphs, particularly on the continuity and switching property of p-Laplacian spectrum of signed graphs.In Section 3, we present the upper bounds of strong and weak nodal domains for p-Laplacians on signed graphs, and discuss the related nodal domain properties on forests.In Section 4, we show multi-way Cheeger inequalities related to strong nodal domains involving p-Laplacians on signed graphs.In Section 5, we establish a nonlinear Perron-Frobenius theorem for the largest eigenvalue of the p-Laplacian on antibalanced graphs.In Section 6, we develop the interlacing theorem which is a signed version of Weyl-like inequalities proposed in [24].Finally, we show lower bound estimates for the number of strong nodal domains in Section 7.

Preliminaries
To explain the interesting story clearly, let us present our setting and notations in this section.Let G = (V, E) be a finite graph with a positive edge measure w : In this paper, we work on a signed graph Γ = (G, σ) with an additional signature σ : E → {−1, 1}.We use C(V ) to denote the set of all the real functions on V , and we always identify C(V ) with R n , i.e., C(V ) ∼ = R n .We denote w({x, y}), κ(x), µ(x) and σ({xy}) by w xy , κ x , µ x and σ xy for simplicity.
In this paper, we assume p ≥ 1.Let Φ p : R → R be defined as Φ p (t) = |t| p−2 t if t = 0 and Φ p (t) = 0 if t = 0. We also write x ∼ y when {x, y} ∈ E.
For p > 1, the signed p-Laplacian ∆ σ p : C(V ) → C(V ) is defined [3,24] by A nonzero function f : V → R is an eigenfunction of ∆ σ p associated with the eigenvalue λ if the following identity holds ∆ σ p f (x) = λµ x Φ p (f (x)), ∀x ∈ V.The signed 1-Laplacian ∆ σ 1 [13,14] is a set-valued map defined by In this paper, we always use Sgn to denote the above set-valued sign function.And we use sgn to denote the usual sign function as follows For a nonzero function f : V → R, we say that it is an eigenfunction of ∆ σ holds in the language of Minkowski sum of convex sets.We will also discuss eigenfunctions with minimal supports (see Theorem 4 in the next section).
Definition 2.1.For any function g : V → R, define supp(g) := {x ∈ V : g(x) = 0}.Let f be an eigenfunction of ∆ σ p corresponding to λ.We say f has minimal support if for any eigenfunction g of ∆ σ p corresponding to λ with supp(g) ⊂ supp(f ), we must have supp(g) = supp(f ).Next, we define balanced and antibalanced graphs.The definition given below is equivalent to the original one by Harary [32] due to Zaslavsky's switching lemma [54].
Definition 2.4.A balanced (resp., antibalanced) graph is a signed graph which is switching equivalent to a graph whose edges are all positive (resp., negative).Remark 2.1.For κ = 0, ∆ σ p is the usual p-Laplacian on signed graphs.For σ ≡ +1, ∆ σ p is nothing but the usual p-Schrödinger operator on graphs.It is known that the graph p-Schrödinger eigenvalue problem covers the Dirichlet p-Laplacian eigenvalue problem on graphs, see, e.g., [34].
For p = 2, ∆ σ 2 reduces to an arbitrary symmetric matrix by taking certain parameters w, σ, µ and κ.
Before giving the following definition, we recall that a set S in a Banach space is centrally symmetric if S = −S where −S := {−x : x ∈ S}.Definition 2.5 (index).The index (or Krasnoselskii genus) of a compact centrally symmetric set S in a Banach space is defined by The following proposition can be found in [48,Proposition 5.2].
Proposition 2.1.For any bounded centrally symmetric neighborhood Ω of the origin in R m , we have γ(∂Ω) = m.
For convenience, we omit the symbol V if no confusion arises, e.g. S p := S p (V ), F k (S p ) := F k (S p (V )).Denote by x∈V µ x |f (x)| p the p-Rayleigh quotient.The Lusternik-Schnirelman theory allows us to define a sequence of variational eigenvalues of ∆ σ p : Moreover, each variational eigenvalue is an eigenvalue of ∆ σ p .It is worth noting that there does exist graphs with non-variational eigenvalues, see [3,Theorem 6].It is proved in [24,Theorem 3.7] that forests admit only variational eigenvalues.Definition 2.6 (eigenspace).The eigenspace X λ (∆ σ p ) of ∆ σ p corresponding to an eigenvalue λ is the subset of S p consists of the all eigenfunctions corresponding to λ.
The multiplicity of an eigenvalue λ of ∆ σ p is defined to be γ(X λ (∆ σ p )), and we shall denote it by multi(λ(∆ σ p )).In this paper, we write λ k to denote λ k (∆ σ p ), if it is clear.
Definition 2.7 (variational multiplicity).For a variational eigenvalue λ of ∆ σ p , its variational multiplicity is defined as the number of times λ appears in the sequence of variational eigenvalues.We will denote it by multi v (λ(∆ σ p )).
It is known that for any variational eigenvalue, its variational multiplicity is always less than or equal to its multiplicity [48,Lemma 5.6].Definition 2.8 (nodal domains, Definitions 3.1-3.4 in [31]).Let Γ = (G, σ) be a signed graph and f : V → R be a function.A sequence {x i } k i=1 of vertices is called a strong nodal domain walk of f if x i ∼ x i+1 and f (x i )σ x i x i+1 f (x i+1 ) > 0 for each i = 1, 2, . . ., k − 1.
A sequence {x i } k i=1 , k ≥ 2 of vertices is called a weak nodal domain walk of f if for any two consecutive non-zeros x i and x j of f , i.e., f (x i ) = 0, f (x j ) = 0, and f (x ℓ ) = 0 for any i < ℓ < j, it holds that We remark that every walk containing at most 1 non-zeros of f is a weak nodal domain walk.
Let Ω = {x ∈ V : f (x) = 0} be the set of non-zeros of f .We denote by {W i } n W i=1 the equivalence classes of the relation W ∼ on Ω.We call the induced subgraph of each set W 0 i := W i ∪ {v ∈ V : there exists a weak nodal domain walk from v to some vertex in W i } a weak nodal domain of the function f .We denote the number n W of weak nodal domains of f by W(f ).
Note that {W i } n W i=1 is a partition of Ω := {x ∈ V : f (x) = 0}.And W 0 i is obtained by adding some zeros to W i .
Next, we give two examples to illustrate this definition.
In Table 1 Example 2.2.We consider a signed star graph Γ = (G, σ) depicted in Figure 2 and its signed Laplacian matrix: Figure 2: The signed star graph.
The eigenvalues of M are λ 1 = 0 < λ 2 = λ 3 = λ 4 = 1 < λ 5 = 5.We consider the eigenfunction f = (0, 1, 1, −1, −1) corresponding to λ 2 .It is direct to check that there are 4 strong nodal domains of f .Next, we investigate the weak nodal domains.Observe that 3 → 1 → 2 and 4 → 1 → 5 are both weak nodal domain walks of f .And there are no weak nodal domain walks between {2, 3} and {4, 5}.Using the notation of Definition 2.8, we have W 1 = {2, 3} and W 2 = {4, 5}.Furthermore, we have W 0 1 = {1, 2, 3} and W 0 2 = {1, 4, 5}.That is, f has two weak nodal domains.Now, we recall two propositions from [31, Propostions 3.16 and 3.17] which will be useful later in the proof of Theorem 3. Proposition 2.2.Let {D i } q i=1 be the all weak nodal domains of a non-zero function f on a signed graph Γ = (G, σ).Let G D = (V D , E D ) be the graph given by , and where D i ∼ D j means that there exist x ∈ D i and y ∈ D j such that x ∼ y.Then, if the graph G is connected, so does the graph G D .
Proposition 2.3.Let f be a non-zero function on a signed graph Γ = (G, σ).Then for any three weak nodal domain Remark 2.2.Another way to study the discrete nodal domains is to consider the edges instead of vertices.Given a function f , define two edge sets Then the number of strong nodal domains of a function f is equal to the number of connected components of the graph [43] proved the upper bound of the signed strong nodal domains by considering the graph Γ ′ .When f is a generic eigenfunction, i.e., f is simple and non-zero on every vertex, the set E − is regarded as the nodal set of f , and the cardinality of E − is called the nodal count of f .The properties of nodal count have been studied in, e.g., [2,7,8,18].The nodal count of signed Laplacian plays an important role in the extension of the Nodal Universality Conjecture from quantum graphs [1] to discrete graphs [2].
We use S(f ) (resp., W(f )) to denote the number of strong (resp., weak) nodal domains of f with respect to (G, −σ).
The perturbation theory plays an important role in studying of the properties of linear operators [41].The following proposition is about the perturbation theory of eigenvalues of p-Laplacian.To state the proposition, we first recall the definition of upper hemi-continuity of set-valued maps.Definition 2.9.Let X and Y be metric spaces.A set-valued map F : X → P(Y ), where P(Y ) stands for the collection of all subsets of Y , is called upper hemi-continuous at x ∈ X if for any neighborhood U of F (x) in Y , there exists η > 0, such that for any

It is said to be upper hemi-continuous if it is upper hemi-continuous at any point of X.
Proposition 2.4.The k-th variational eigenvalue is continuous with respect to Moreover, the multiplicity and variational multiplicity of the k-th variational eigenvalue are both upper semi-continuous with respect to (w, κ, µ) and the corresponding eigenspace is upper hemi-continuous with respect to (w, κ, µ).In particular, the set of the parameters (w, κ, µ) it is easy to show that the k-th variational eigenvalue is continuous with respect to (w, κ, µ).First, we prove the upper semi-continuity of the variational multiplicity.Let r be the variational Without loss of generality, we assume that ).By continuity, the above two inequalities hold in an open neighborhood U of (w 0 , κ 0 , µ 0 ).Therefore, the variational multiplicity of λ k (∆ σ p [w, κ, µ]) with (w, κ, µ) ∈ U is equal to or less than r.This proves the upper semi-continuity of the variational multiplicity.
Next, we prove the upper semi-continuity of the multiplicity.Let X k (w, κ, µ) ⊂ S p be the collection of all normalized eigenfunctions corresponding to the k-th variational eigenvalue of ∆ σ p with the parameter (w, κ, µ).We first verify that the eigenspace X k (w, κ, µ) is upper hemi-continuous with respect to (w, κ, µ).Suppose the contrary, that there exists ǫ 0 > 0 such that there exists a sequence by the compactness, there exists a subsequence, still denoted by we have the eigen-equation By the monotonicity and continuity of the index function γ ([48, Proposition 5.4]), we have )) indicates the multiplicity of the k-th variational eigenvalue of ∆ σ p with the parameter (w, κ, µ).This implies that {(w, κ, µ) Hence the multiplicity of the k-th variational eigenvalue is upper semicontinuous with respect to (w, κ, µ).
In the linear case, we know that if (G, σ) and (G, σ) are switching equivalent with the same edge measure, vertex weight and potential function, then the spectrum of ∆ σ 2 coincides with that of ∆ σ 2 .The following proposition shows this fact still holds for the nonlinear case.Proposition 2.5.Let (G, σ) and (G, σ) be two signed graphs with the same edge measure, vertex weight and potential function.If σ is switching equivalent to σ, then the spectrum of ∆ σ p coincides with the spectrum of ∆ σ p .Moreover, the variational spectra of ∆ σ p and ∆ σ p are the same.
Proof.Suppose σ := σ τ for some switching function τ : V → {−1, +1}.By direct computation, we derive that (λ, f ) is an eigenpair of ∆ σ p if and only if (λ, τ f ) is an eigenpair of ∆ σ τ p .Therefore, the set of eigenvalues of ∆ σ p agrees with the set of eigenvalues of ∆ σ τ p .Note that for any centrally symmetric subset p ) of the eigenfunctions corresponding to the eigenvalue λ of ∆ σ τ p .Hence, the multiplicity of the eigenvalue λ of ∆ σ p coincides with the multiplicity of the eigenvalue λ of ∆ σ τ p .In summary, we obtain that the spectra of ∆ σ p and ∆ σ τ p coincide.Finally, we focus on the variational eigenvalues.It is direct to check that γ(A) = γ(τ • A) for any centrally symmetric subset A. And for any minimizing set A with respect to

Nodal domain theorems
In this section, we prove nodal domain theorems for p-Laplacians on signed graphs and discuss several applications.Let Γ = (G, σ) be a signed graph with G = (V, E), and let be the variational eigenvalues of ∆ σ p .For ease of notation, we denote n = |V |.For any eigenfunction f corresponding to λ, we prove the following upper bounds for the quantities S(f ), W(f ), S(f ) and W(f ).
where c is the number of connected components of G.
and the corresponding eigenfunction f has minimal support, then we have In addition, when p = 1, and f has minimal support, we further have that S(f ) = 1.Moreover, when the graph is balanced, the number of zeros of f is at least k + r − 2.
Let us first remark on the estimates of S(f ) (resp., W(f )), i.e., the number of strong (resp., weak) nodal domains of f with respect to (G, −σ).In the linear case, if f is an eigenfunction of the signed Laplacian ∆ σ 2 corresponding to λ, then it is also an eigenfunction of −∆ σ 2 corresponding to −λ.Since −∆ σ 2 can be considered as a signed Laplacian of the graph (G, −σ) (with a suitable choice of the potential function), the upper bound estimates of S(f ) and W(f ) follows directly from the signed nodal domain theorem [31,Theorem 4.1].However, in the non-linear case, when f is an eigenfunction of ∆ σ p , f may not be an eigenfunction of ∆ −σ p anymore.It is an interesting question to ask whether there are still upper bound estimates of S(f ) and W(f ) or not.Theorem 2, Theorem 3 and Theorem 4 above answer this question positively.Intriguingly, these upper bound estimates will be very useful in the proofs of our later results, including Theorem 5, Theorem 6 and Theorem 9.
Those above upper bounds can be regarded as discrete versions of the Courant's nodal domain theorem [20,21] proved in 1920s.Cheng [16] studied Courant's theorem on Riemannian manifolds.The study of discrete nodal domain theorems for linear Laplacians on graphs dates back to the work of Gantmacher and Krein [30] in 1940s and the work of Fiedler [26][27][28] in 1970s.Van der Holst [51,52] proved that the second eigenfunction f 2 induces 2 strong nodal domains if it has minimal support.Duval and Reiner [25] studied the discrete nodal theorems of higher eigenfunctions.In 2001, Davies, Gladwell, Leydold and Stadler [23] established the discrete nodal domain theorems for generalized Laplacians.There are amount of works about discrete nodal domain theorems for linear Laplacians, see, e.g., [7,9,19,29,39,45,46].The extensions to linear Laplacians on signed graphs have been discussed in [31,38,43], while the extensions to non-linear Laplacians on graphs have been carried out in [15,24,49].
Those above results unify many results on the upper bounds of the number of nodal domains for p-Laplacians on graphs and signed graphs, including [31, Theorem 4.1], [38,Theorem 5.4] for signed graphs, [49, Theorem 3.4 and Theorem 3.5] for graphs.Moreover, the inequality (see [38,Theorem 5.3 ] and [40, where N(f ) stands for the number of connected components of the support of f , becomes a direct consequence of these results, since we have We further point out that Theorem 3 can not hold for the case p = 1, even for balanced signed graphs.A counterexample is given in [15, Example 10].
For the proofs of these theorems, we prepare two lemmas.The first one has been established in [3,38,49].
Lemma 1.Let t, s, a, b be real numbers.Then, we have for p > 1

Moreover, the equality holds if and only if
in both cases.
In the case of p = 1, we have for any z ∈ Sgn(a + b), For any function g : V → R, we define g p p = x∈V |g(x)| p µ x for p ≥ 1.We will use the notation i =j := i j:j =i for simplicity.Lemma 2. For p ≥ 1, let f be an eigenfunction of ∆ σ p corresponding to an eigenvalue λ.Set Let X be the linear function-space spanned by f 1 , . . ., f m where Then, for any g = m i=1 t i f i ∈ X \ 0, we have where Proof.We first compute for any p ≥ 1 that {x,y}∈E We next deal with the case p > 1. Employing the eigen-equation, we have for each i ∈ {1, . . ., m} Consequently, we obtain (5) Combining (3) and ( 5), we get {x,y}∈E where This completes the proof for the case p > 1.
Finally, we discuss the case p = 1.By definition, we have for any x ∈ V .Hence, there exist x , for any x ∈ V .For any i ∈ {1, . . ., m}, we compute Consequently, we derive Combining ( 3) and ( 7) yields {x,y}∈E where This completes the proof for the case p = 1.
We are now well-prepared for the proof of Theorem 1.
Proof of Theorem 1.By definition, we have W(f ) ≤ S(f ).Next, we prove S(f ) ≤ k.
Suppose that f has m strong nodal domains on Γ = (G, σ) which are denoted by V 1 , . . ., V m .Consider the linear function-space X spanned by f 1 , . . ., f m , where f i is defined by Since V 1 , . . ., V m are pairwise disjoint, we have dim X = m.Then we can use Proposition 2.1 to get γ(X ∩ S p ) = m.
We claim that R σ p (g) ≤ λ for any g = m i=1 t i f i ∈ X \ 0. Indeed, we have by Lemma 2, For any i = j, x ∈ V i and y ∈ V j , we take a = f i (x), b = −σ xy f j (y), t = t i and s = t j .Because x and y lie in different strong nodal domains, we have ab = −f i (x)σ xy f j (y) > 0. Then we use Lemma 1 to get G ij (x, y) ≤ 0. That is, we have R σ p (g) ≤ λ.By definition, we have This implies m ≤ k.
In order to prove the upper bound of S(f ) in Theorem 2, we recall the following lemma from [44, Proposition 4.2.20].
is the projection operator onto Y , and A is a closed centrally symmetric subset with Proof of Theorem 2. By definition, we have W(f ) ≤ S(f ).Next, we prove S(f ) ≤ n − k.
As above, we suppose that f has m strong nodal domains on Γ ′ = (G, −σ) which are denoted by V 1 , . . ., V m .Let X be the linear function-space spanned by f 1 , . . ., f m , where f i is defined as follows We first prove that R σ p (g) ≥ λ for any g = n i=1 t i f i ∈ X \ 0. Indeed, we have by Lemma 2, For any i = j, x ∈ V i and y ∈ V j , we take a = f i (x), b = −σ xy f j (y) and t = t i , s = t j .Because x and y lie in different strong nodal domains on Γ = (G, −σ), we have by definition ab = −f i (x)σ xy f j (y) < 0.
Then we use Lemma 1 to get G ij (x, y) ≥ 0. That is, we have R σ p (g) ≥ λ.Notice that, by Lemma 3, X ′ ∩ X = ∅ for any X ′ ∈ F n−m+1 (S p ). Then we have by definition This completes the proof.
To show the upper bounds of W(f ) and W(f ) in Theorem 3, we prepare the following two lemmas: The first one is a reformulation of a related result by Hein and Tudisco [ Then A * contains at least one critical point of R σ p corresponding to λ k .
Proof.The proof follows the same line of that of [49,Lemma 2.3], with the only difference being that the deformation lemma is used to construct an odd continuous map to deform the minimizing set A * .
Lemma 5.For p ≥ 1 and k ≥ 1, let X be a linear subspace of dimension n − k + 1 such that Then X ∩ S p contains as least one critical point of R σ p corresponding to λ k .
Proof.We first concentrate on the case of p > 1. Suppose the contrary, that X ∩ S p has no critical points of R σ p corresponding to λ k .Let K λ k (R σ p ) be the set consists of all critical points in S p of R σ p corresponding to λ k .By definition, we know K λ k (R σ p ) is closed.By assumption, we have Then there exists a neighborhood of where ǫ > 0 is sufficiently small.In particular, we have Let A be a minimizing set corresponding to λ k .We have γ(A) ≥ k.Since θ is an odd homeomorphism, the inverse map θ −1 is odd continuous.By the continuity property of the index function γ, we have γ(θ −1 (A)) ≥ k.So, by the intersection property of the index function γ (see also Lemma 3), Then, we obtain which is a contradiction.For the case of p = 1, we consider the restriction R where TV(g) := {x,y}∈E w xy |g(x) − σ xy g(y)| + κ x |g(x)|.According to the facts g, ∆ σ 1 g = TV(g) and g, µSgn(g) = g 1 , we have g, ∂R σ 1 (g) = 0, i.e., g, h = 0 for any h ∈ ∂R σ 1 (g).So, we have That is, the set of critical points of R σ 1 with l 2 -norm one coincide with the that of the restriction R σ 1 | S 2 .We then apply [13, Theorem 3.1, Remarks 3.3 and 3.4] to deduce that there is an odd homeomorphism θ : where ǫ > 0 is sufficiently small.Let η : S 1 → S 2 be an odd homeomorphism defined as η(f ) = f / f 2 .Then, along the line of the proof for the case of p > 1, we derive for a minimizing set A ⊂ S 1 corresponding to λ k that, which is a contradiction.
Proof of Theorem 3: Upper bound of W(f ).Suppose f has m weak nodal domains which are denoted by U 1 , . . ., U m .Let W 1 , . . ., W c be the c connected components of the graph.Then, for any i ∈ {1, . . ., m}, there exists a unique l ∈ {1, . . ., c} such that U i ⊂ W l .For l = 1, . . ., c, We denote by the index set corresponding to W l .Then, we have c l=1 I l = {1, . . ., m}.We prove that by contradiction.Assume m ≥ k + c.Let X be the linear function-space spanned by f | U 1 , . . ., f | Um where f | U i = f on U i and f | U i = 0 on V \ U i for any 1 ≤ i ≤ m.Let X ′ be the linear function-space spanned by f | W 1 , . . ., f | Wc where f | W j = f on W j and f | W j = 0 on V \ W j for any 1 ≤ j ≤ c.Similarly as the proof of Theorem 1, we drive from Lemma 1 and Lemma 2 that By definition, we have f According to the definition of variational eigenvalues, there holds So we have max Let U i and U j be two adjacent weak nodal domains.If there exist x 0 ∈ U i and y 0 ∈ U j such that {x 0 , y 0 } ∈ E, f (x 0 ) = 0 and f (y 0 ) = 0, then we derive from the condition (1) in Lemma 1 that t i = t j .If, otherwise, there exist x 0 ∈ U i and y 0 ∈ U j such that {x 0 , y 0 } ∈ E and f (x 0 ) = 0, f (y 0 ) = 0 or f (x 0 ) = 0, f (y 0 ) = 0, then we claim t i = t j still holds.Without loss of generality, we assume f (x 0 ) = 0 and f (y 0 ) = 0.
Indeed, since f and g are eigenfunctions, we have y∼x w x 0 y Φ p (σ x 0 y f (y)) = 0, and y∼x w x 0 y Φ p (σ x 0 y g(y)) = 0.
We derive from Proposition 2.3 that every y ∼ x 0 lies in either U i or U j .In fact, if there exists y ∼ x 0 such that y ∈ U k for some k = i, j, then we have x 0 ∈ U i ∩ U j ∩ U k by definition of weak nodal domains and the fact f (x 0 ) = 0.This contradicts to Proposition 2.3.From the equalities in (9), we obtain and hence, (Φ p (t i ) − Φ p (t j )) By definition of weak nodal domain walk, for any y, y ′ ∈ U j with {x 0 , y}, {x 0 , y ′ } ∈ E, we have (σ x 0 y f (y)) • (σ x 0 y ′ f (y ′ )) = f (y)σ yx 0 σ x 0 y ′ f (y ′ ) ≥ 0 and f (y 0 ) = 0, which implies that Thus, we derive from (10) that Φ(t i ) − Φ(t j ) = 0, which yields t i = t j .
In conclusion, we have t i = t j whenever U i and U j are adjacent.Thus, in each connected component W l , we use Proposition 2.2 to get t i = t j whenever i, j ∈ I l .But this implies g ∈ X ′ \ 0, which is a contradiction with g ∈ Y .This completes the proof of W(f ) ≤ k + c − 1.
Next, we prove the upper bound of W(f ).
Proof of Theorem 3: Upper bound of W(f ).Suppose f has m weak nodal domains which are denoted by U 1 , . . ., U m with respect to the opposite signed graph (G, −σ).
Suppose, to the contrary, that m ≥ n − k − r + c + 2. Let {W i } c i=1 be the connected components of G.For any 1 ≤ i ≤ m, let f | U i be the function that equals f on U i and zero on V \ U i .Define X to be the linear function-space spanned by f | U 1 , . . ., f | U m .For any 1 ≤ j ≤ c, let f | W j be the function that equals f on W j and equals zero on V \ W j .Define X ′ to be the linear function-space spanned by f | W 1 , . . ., f | W c .As above, X ′ is a linear subspace of X and we can have a decomposition Following the same line of the proof of Theorem 2, we drive from Lemma 1 and Lemma 2 that Observe by Lemma 3 that A ∩ Y = ∅ for any A ∈ F k+r−1 (S p ). Then we prove that So the above inequalities hold with equalities.In particular, min Then, Lemma 5 implies that there exists an eigenfunction Along the same line of the proof for W(f ) ≤ k + c − 1, we get a contradiction that the nonzero function g belongs to both X ′ and Y , which completes the proof.
In the following, we prove Theorem 4. For the p = 1 part of Theorem 4, we show the following lemma.
Lemma 6 (localization property of 1-Laplacian).Let (λ, f ) be an eigenpair of ∆ σ 1 .Then, for any strong nodal domain U of f , and any c ≥ 0 such that {x ∈ U : f (x) > c} or {x ∈ U : f (x) < −c} is non-empty, both f | U and 1 {x∈U :f (x)>c} − 1 {x∈U :f (x)<−c} are eigenfunctions corresponding to the same eigenvalue λ of ∆ σ 1 .In addition, if f has minimal support, then f has only one strong nodal domain, denoted by U , and f must be in the form of t(1 A − 1 B ) for some t = 0 and some disjoint subsets A, B with for any x, y ∈ V , any c ≥ 0 and any strong nodal domain U of f .It means that as a set-valued map, x) for any x ∈ V .Since f is an eigenfunction corresponding to an eigenvalue λ of ∆ σ 1 , we have the differential inclusion for any x ∈ V .That is, both f | U and f U,c are eigenfunctions corresponding to λ.Now, we further assume that f has minimal support.Then, by the localization property proved above, f has only one strong nodal domain, denoted by U .Suppose, to the contrary, that f is not in the form of t(1 A − 1 B ). Then there exists c > 0 such that the support of f U,c is a nonempty proper subset of U .So, we construct an eigenfunction f U,c corresponding to the eigenvalue λ, but its support is a proper subset of the support of f , which leads to a contradiction with the minimal support assumption on f .Therefore, we have shown that f is in the form of t(1 A − 1 B ), and its strong nodal domain U is the disjoint union of A and B. Clearly, for any g whose support is included in U , if g is also an eigenfunction corresponding to the eigenvalue λ, g = t ′ (1 A ′ − 1 B ′ ) for some t ′ = 0 and some disjoint subsets A ′ and B ′ with } is a finite set, and its index is one.
Proof of Theorem 4. Recall we assume that f has minimal support.
We first prove that S(f ) ≤ k.Let {V i } m i=1 be the strong nodal domains of f on Γ = (G, σ).We prove it by contradiction.Assume m > k.Consider two linear spaces X and X ′ defined as follows By Proposition 2.1, we have γ(X ∩ S p ) = m > k and γ(X ′ ∩ S p ) = m − 1 ≥ k.By definition of variational eigenvalues, we get Therefore, all the inequalities above are equalities.In particular, X ′ ∩ S p is a minimizing set.By Lemma 4, there exists an eigenfunction g 0 = m−1 i=1 b i f | V i corresponding to λ, which contradicts to the fact that f has minimal support.This proves m ≤ k.
Next, we prove S(f ) ≤ n − k − r + 2. Let {V i } m i=1 be the strong nodal domains of f with respect to the opposite signed graph (G, −σ).We prove it by contradiction.Assume m > n − k − r + 2. Consider two linear spaces X and X ′ defined as By the proof of Theorem 2, we have Then we have Therefore, all the inequalities above are equalities.Next, by Lemma 5, X ′ ∩ S p contains a critical point of R σ p corresponding to λ k .That is, there exists an eigenfunction g = m−1 i=1 b i f | V i ∈ X ′ \ 0 corresponding to the eigenvalue λ k , which contradicts to the fact that f has minimal support.This shows m ≤ n − k − r + 2.
In the particular case of p = 1, we actually have S(f ) = 1 by Lemma 6.Moreover, we can assume without loss of generality that f = 1 A − 1 B for disjoint subsets A and B, where A ∪ B is the strong nodal domain of f .When the graph is balanced, we obtain by the definition of strong nodal domains that S(f Consequently, the number of zeros of f is at least k + r − 2. Next, we present two important applications of the upper bounds for S(f ), S(f ), W(f ) and W(f ) in Theorem 1, Theorem 2, and Theorem 3. The estimates of the quantity S(f ) + S(f ) for an eigenfunction f will play an essential role.
Theorem 5. Let Γ = (G, σ) be a signed graph with G = (V, E).Let f be an eigenfunction corresponding to a non-variational eigenvalue.If |E| < |V |, then f must have zeros.
We emphasize that the graph G = (V, E) in the above theorem is allowed to be disconnected.
Proof.We prove it by contradiction.We assume that f is non-zero on all vertices.Define By assumption, we have By definition of strong nodal domains, we have where n = |V |.Let k be the index such that λ k < λ < λ k+1 , where λ is the eigenvalue to f .Then, Theorems 1 and 2 tell that S(f ) ≤ k and S(f Combining the above inequalities, we have which is a contradiction. On a forest G, Theorem 5 implies that any eigenvalue λ with an everywhere non-zero eigenfunction f must be a variational eigenvalue.This can be strengthened as follows.Theorem 6 below has been obtained in [24,Theorem 3.8].We provide here an alternative simple proof using the estimates of nodal domains and anti-nodal domains.Theorem 6.Let G = (V, E) be a forest with c connected components and f be an everywhere non-zero eigenfunction corresponding to an eigenvalue λ.Then λ is a variational eigenvalue with variational multiplicity c and f has exactly k + c − 1 strong nodal domains.
This can be regarded as a non-linear version of the results on the linear Laplacian [7,9,27].
Proof.Since G is a forest, we have |V | − |E| = c > 0. By Theorem 5 and the assumption that f is non-zero on every vertex, λ is a variational eigenvalue.We assume that λ = λ k and We define By definition of strong nodal domains, we have where n = |V |.This yields We first prove r ≤ c.Since f is non-zero on every vertex, we can use Theorem 3 to get When | for short.We further have the following notations for boundary measure and volume: For ease of notation, we denote n = |V |.
Definition 4.1.[4, Definition 3.2] For any integer 1 ≤ k ≤ n, the k-way signed Cheeger constant h σ k of a signed graph Γ = (G, σ) is defined as where and the minimum is taken over all possible k-sub-bipartitions, i.e., ( It is direct to check the following monotonicity of the multi-way singed Cheeger constants.For the readers' convenience, we provide a proof below.

Lemma 7 (Monotonicity). For any integer
By definition, we have Next, by direct computation, we get So this implies Remark 4.1.The above signed Cheeger constants on signed graphs can be considered as an optimization of a mixture of isoperimetric constant and the so-called frustration index.The frustration index ι σ (Ω) of a subset Ω ⊂ V measures how far the signature on Ω is from being balanced.It is defined as By switching, we see ι σ (Ω) = 0 if and only if the signature restricting to the subgraph induced by Ω is balanced.Indeed, the k-th signed Cheeger inequality can be reformulated as [42] h σ k := min This can be verified using the one-to-one correspondence between the function τ : Ω i → {±1} and the bipartition Notice that h σ k reduces to the classical k-th Cheeger constant when Γ = (G, σ) is balanced, since ι σ (Ω i ) vanishes for any subset Ω i .Theorem 8.For any p ≥ 1 and any k ∈ {1, . . ., n}, the k-th variational eigenvalue λ k (∆ σ p ) satisfies where C := max x∈V y wxy µx and m is the number of strong nodal domains of an eigenfunction corresponding to λ k (∆ σ p ).
This theorem can be regarded as a signed version of [49, Theorem 5.1], which is an extension of previous works [3,10,13,22].
Before proving this theorem, we first show an elementary inequality.Proof.Without loss of generality, we can assume ab = 0. We consider the case of σ ab = −1 below.
The proof for the case of σ ab = 1 can be done similarly.
If ab > 0, we assume a > 0 and b > 0 without loss of generality.Then we get By the convexity of f (x) = |x| p , we have f If, otherwise, ab < 0, we assume a > 0, b < 0, and a = −kb with k > 1 without loss of generality.Then we get |a − σ ab b| p = |a + b| p = |k − 1| p |b| p .
By the convexity of the following function This completes the proof of the case σ ab = −1.
Proof of Theorem 8. Observe that for any k-sub-bipartitions where We first show the upper bound estimate of λ k .By abuse of notation, we use we derive by Lemma 8 that Therefore, we compute By definition of the variational eigenvalue λ k , we obtain λ k ≤ 2 p−1 h σ k .Next, we prove the lower bound estimate of λ k .Let f be an eigenfunction corresponding to λ k , and let V 1 , • • • , V m be the strong nodal domains of f .By the proof of Theorem 1, we have where f i equals f on V i and equals zero otherwise.We prove two claims.Claim 1.For any i = 1, . . ., m, we denote by f p i : V → R the function x → |f i (x)| p sgn(f i (x)).Then we have Indeed, by [3, Lemma 3], we have Following the proof of [49, Lemma 5.2], we obtain This proves Claim 1. Claim 2. There exist For any t ≥ 0, define and a function f t : V → R as follows otherwise.
Then, we have Note that the function f p i is defined as in Claim 1.So by direct calculation, we have Therefore, there exists t 0 ≥ 0 such that where U 2i−1 := V t 0 + (f i ) and U 2i := V t 0 − (f i ).This completes the proof of Claim 2. Combining the above two claims, we get In consequence, Moreover, suppose Γ has k + l connected components denoted by Γ 1 , . . ., Γ k+l , in which Γ 1 , . . ., Γ k are balanced, while Γ k+1 , . . ., Γ k+l are not balanced.Then, the smallest positive eigenvalue of the p-Laplacian coincides with the (k + 1)-th variational eigenvalue, which can be expressed as follows where λ s (∆ σ p | Γ i ) indicates the s-th variational eigenvalue of the p-Laplacian restricted on Γ i .
Proof.We first assume Γ has exactly k balanced connected components.Then by [4, Proposition 3.2], we have Denote by m the number of balanced connected components of Γ.Along the same line of the above arguments, we derive that Comparing with our assumption, we have m = k.
Next, we prove (16).It is direct to check that the eigenvalue of ∆ σ p on Γ is the multiset-sum of the eigenvalue of ∆ σ p on Γ i for i = 1, • • • , k + l, i.e., {λ : λ is an eigenvalue of ∆ σ p on Γ} = ⊕ k+l i=1 {λ : λ is an eigenvalue of ∆ σ p on Γ i }.
For particular cases, the variational eigenvalues of the 1-Laplacian might coincide with the signed Cheeger constants.
Proof.Let f 1 be an eigenfunction corresponding to λ 1 (∆ σ 1 ).Setting p = 1 and k = 1 in Theorem 8 leads to Theorem 9. Assume that p > 1.Let Γ = (G, σ) be a connected signed graph where σ ≡ −1, G = (V, E) and |V | = n.For any eigenfunction f corresponding to the n-th variational eigenvalue λ n of ∆ σ p , we have the following properties: (i) f is either strictly positive or strictly negative, i.e., either f (x) > 0 for any x ∈ V or f (x) < 0 for any x ∈ V ; (ii) For any other eigenfunction g corresponding to λ n , there exists a constant c ∈ R \ {0} such that g = cf ; (iii) If g is an eigenfunction corresponding to an eigenvalue λ, and g(x) > 0 for any x ∈ V or g(x) < 0 for any x ∈ V , then λ = λ n .
Let us remark that the Perron-Frobenius theorem above does not hold for the case of p = 1.Indeed, according to Theorem 4, there exists an eigenfunction f corresponding to λ n of ∆ σ 1 such that S(f ) = 1.However, if Theorem 9 were true for p = 1, we would have S(f ) = n for any eigenfunction corresponding to λ n of ∆ σ 1 , which is a contradiction.
Proof of Theorem 9. (i) Since λ n is the n-th variational eigenvalue, Theorem 3 implies W(f ) ≤ 1.By definition of weak nodal domains, we have f (x) ≥ 0 for any x ∈ V or f (x) ≤ 0 for any x ∈ V .We can assume f (x) ≥ 0 for any x ∈ V , since otherwise, we can consider the eigenfunction −f .If f (x) = 0 for some x ∈ V , we have by the eigen-equation that Since σ ≡ −1, we obtain y∼x w xy Φ p (f (y)) = 0.Because f (y) is non-negative for all y ∈ V , we have f (y) = 0 for all y with y ∼ x.By the connectedness of G, we have f ≡ 0. This contradicts to the assumption that f is an eigenfunction of λ n .Thus, we get f (x) > 0 for any x ∈ V .
(ii) Suppose that g is an eigenfunction corresponding to λ n .Without loss of generality, we can assume g(x) > 0 for any x ∈ V .By definition, we have for any Multiplying (20) by f (x) − |g(x)| p Φp(f (x)) , and (21) by g(x) − |f (x)| p Φp(g(x)) , we derive Summing ( 22) and ( 23) over all x ∈ V , we get where We apply Lemma 1 by setting a = f (x), b = f (y), ta = g(x) and sb = g(y) to derive that each summand in R(f, g) is non-positive.Similarly, we have each summand in R(g, f ) is also non-positive.
Therefore, the identity (24) implies that every summand of R(f, g) and R(g, f ) equals zero.By the equality condition (1) in Lemma 1, we have for any {x, y} ∈ E that Since G is connected, we drive that g is proportional to f .This concludes the proof of (ii).
(iii) If g is an eigenfunction corresponding to λ and g(x) > 0 for any x ∈ V .By definition, we have As above, we multiply (25) by f (x) − |g(x)| p Φp(f (x)) and ( 26) by g(x) − |f (x)| p Φp(g(x)) , and sum them over all x ∈ V .Then, we obtain We can choose sufficiently small ǫ > 0 such that f (x) − ǫg(x) > 0 for any x ∈ V .So without loss of generality, we can assume |f (x)| p − |g(x)| p > 0 for any x ∈ V .If λ < λ n , then the right hand side of ( 27) is strictly positive and the left hand side of ( 27) is non-positive.This is a contradiction.The proof of λ = λ n is then completed.
Notice that a connected bipartite graph with σ ≡ 1 is both balanced and antibalanced.Hence, our Theorem 9 covers the conclusion of [24,Theorem 4.4] and [34,Theorem 1.2].Next, we use Theorem 9 to derive the following results.
Theorem 10.Let Γ = (G, σ) be a connected antibalanced signed graph and {λ i } n i=1 be the variational eigenvalues of ∆ σ p with p > 1.Then we have λ n−1 < λ n and there are no other eigenvalues between λ n−1 and λ n .
We prove the theorem by contradiction.Assume that λ is an eigenvalue satisfying λ n−1 < λ < λ n and f is an eigenfunction corresponding to λ.By Theorem 2, we get S(f ) ≤ 1.Then by definition of S, we have f ≥ 0 on every vertex or f ≤ 0 on every vertex.We assume f ≥ 0 on every vertex and the case that f ≤ 0 on every vertex can be proved similarly.If f is zero on some x ∈ V , we have by the eigen-equation that y∼x So we have y∼x w xy Φ p (f (y)) = 0.Because f (y) ≥ 0 for any y ∈ V , we obtain f (y) = 0 for any y ∼ x.By the connectedness of Γ, we have f = 0 on all vertices, which can not happen.So f is positive on all vertices.Then, we apply Theorem 9 to get λ = λ n , which leads to a contradiction.
Using again the fact that a bipartite graph with σ ≡ 1 is antibalanced, we derive from Theorem 10 the following corollary.
Corollary 5.1.For any connected bipartite graph, there are no eigenvalues between the largest and the second largest variational eigenvalues of the corresponding p-Laplacian with p > 1.

Interlacing theorems
When one wants to understand a quantitative property of a graph, it is natural to investigate how this quantity changes under modifying the graph via deleting vertices or edges.
In this section, for an eigenpair (λ, f ) of ∆ σ p with p > 1, we give a way to modify a signed graph to a forest T such that (λ, f | T ) is again an eigenpair of T .We estimate how the eigenvalue changes in each step.This leads to a nonlinear version of the Cauchy Interlacing Theorem.The theorems in this section are signed versions of the theorems in [24, Section 5].Those interlacing theorems will be useful for the lower bound estimates of S(f ) in the next section.

Removing an edge
Consider a signed graph Γ = (G, σ), where G = (V, E), with an edge measure w, a vertex weight µ, and a potential function κ.Let f ∈ C(V ) be a function and {x 0 , y 0 } ∈ E be an edge such that f (x 0 )f (y 0 ) = 0. We define a new signed graph with an edge measure w ′ , a vertex weight µ ′ and a potential function κ ′ defined as follows: w ′ xy = w xy for any {x, y} ∈ E ′ , µ ′ x = µ x for any x ∈ V , and Then, the corresponding p-Laplacian with p > 1 of the new signed graph Γ ′ is given by It is direct to check that the above choices of w ′ , µ ′ and κ ′ lead to the following property: If f ∈ C(V ) is an eigenfunction corresponding to an eigenvalue λ of the p-Laplacian ∆ σ p with p > 1, then f is still an eigenfunction of ∆ σ ′ p corresponding to λ.Let R σ ′ p be the Rayleigh quotient of ∆ σ ′ p defined as  (i) if f (x 0 )σ x 0 y 0 f (y 0 ) < 0, then η k−1 ≤ λ k ≤ η k for any 1 < k ≤ n; (ii) if f (x 0 )σ x 0 y 0 f (y 0 ) > 0, then η k ≤ λ k ≤ η k+1 for any 1 ≤ k < n.
Lemma 10.Consider a signed graph Γ = (G, σ) where G = (V, E) and a given vertex x 0 ∈ V .Let ∆ σ p be the corresponding p-Laplacian with p > 1, and Γ ′ = (G ′ , σ), ∆ σ ′ p be defined as above.Denote by λ k and η k the k-th variational eigenvalues of ∆ σ p and ∆ σ ′ p , respectively.Then we have Proof.Define S ′ p = {g : . By definition, we have A k ∈ F k (S p ), and This concludes the proof of the first inequality.Let A k+1 ∈ F k+1 (S p ) be a set such that λ k+1 = max g∈A k+1 R σ p (g). Define This concludes the proof of the second inequality.
We can use Lemma 10 iteratively to get the following theorem.
Combining ( 35) and ( 36), we have This implies The last inequality is because of |E z | ≥ z(f ).Then we complete the proof.

Proof of Theorem 13 (i).
First, since Γ ′ is obtained by removing all zero vertices of f from Γ, we can define a new p-Laplacian on Γ ′ as (32) denoted by ∆ σ ′ p .Next, we remove all the edges in E ′ f − of f on Γ ′ one by one to get the graph Γ ′′ = (G ′′ , σ ′′ ) with G ′′ = (V ′′ , E ′′ ) at end.At each step, we define a new p-Laplacian as in (28).Denote by ∆ σ ′′ p the p-Laplacian on we obtain at end.By Theorem 11 and Theorem 12, we get For any {x, y} ∈ E ′′ , we have f This concludes the proof of (i).
Next, we remove l(G ′ ) edges of Γ ′ to make Γ ′ to be a forest T .Assume that {e i } l(G ′ ) i=1 are all the edges we remove, where e i = {x i , y i }.We define Γ j as the subgraph obtained by removing edges {e i } j i=1 from Γ ′ .At each step, we define a new p-Laplacian on Γ j as in (28) denoted by ∆ σ p,j .Denote by {λ the variational eigenvalues of ∆ σ p,j .

( i )
Define an equivalence relation S ∼ on Ω as follows: For any x, y ∈ Ω, x S ∼ y if and only if x = y or there exists a strong nodal domain walk connecting x and y.We denote by {S i } n S i=1 the equivalence classes of the relation S ∼ on Ω.We call the induced subgraph of each S i a strong nodal domain of the function f .We denote the number n S of strong nodal domains of f by S(f ).

(
ii) Define an equivalence relation W ∼ on Ω as follows: For any x, y ∈ Ω, x W ∼ y if and only if x = y or there exists a weak nodal domain walk connecting x and y.
and hence X ′ is a linear subspace of X.We can have a decomposition X = X ′ Y .Since dim X = m ≥ k + c and dim X ′ = c, we derive dim Y ≥ k, and hence, γ(Y ∩ S p ) ≥ k by Proposition 2.1.

Corollary 4 . 1 .
For any signed graph Γ = (G, σ), we have λ 1 directly from Proposition 2.5 and [13, Theorem 5.15].As a consequence of Proposition 4.1 and Corollary 4.1, we have the following expression of the first positive eigenvalue of the 1-Laplacian.

Table 1 :
Strong and weak nodal domainsIt is worth noting that for the eigenfunction f 3 , vertices 1 and 2 lie in the same weak nodal domain because 1 → 3 → 4 → 5 → 3 → 2 is a weak nodal domain walk.
49, Lemma 2.3]; The second one is a new result for estimating the number of dual nodal domains.It is worth noting that any f ∈ S p is a critical point of R σ p corresponding to λ k if and only if it is an eigenfunction of ∆ σ p corresponding to λ k .
Lemma 4. For p ≥ 1 and k ≥ 1, let A * ∈ F k (S p ) be such that [48,ifold and R σ p is smooth, we can apply[48, Theorem 3.11]to derive that there exists an odd homeomorphism θ : S p → S p with θ({g ∈ S p : R σ p 2 p−1 C p−1 p p (h σ m ) p ≤ λ k .The proof is completed.Proposition 4.1.For any p ≥ 1 and any k ∈ {0, 1, . . ., n}, a signed graph Γ has exactly k balanced connected components if and only if the variational eigenvalues of the p-Laplacian satisfy λ 1 they are all zero.On the other hand, according to [40, Theorem 2.1], the smallest positive eigenvalue of ∆ σ p on Γ is λ k+1 (∆ σ p ).So we have λ k+1 (∆ σ p ) > 0. Conversely, we assume that λ 1