$p$-Laplace Operators for Oriented Hypergraphs

The $p$-Laplacian for graphs, as well as the vertex Laplace operator and the hyperedge Laplace operator for the general setting of oriented hypergraphs, are generalized. In particular, both a vertex $p$-Laplacian and a hyperedge $p$-Laplacian are defined for oriented hypergraphs, for all $p\geq 1$. Several spectral properties of these operators are investigated.


Introduction
Oriented hypergraphs are hypergraphs with the additional structure that each vertex in a hyperedge is either an input, an output or both. They have been introduced in [21], together with two normalized Laplace operators whose spectral properties and possible applications have been investigated also in further works [1,[31][32][33]. Here we generalize the Laplace operators on oriented hypergraphs by introducing, for each p ∈ R ≥1 , two p-Laplacians. While the vertex p-Laplacian is a known operator for graphs (see for instance [3,14,39]), to the best of our knowledge the only edge p-Laplacian for graphs that has been defined is the classical one for p = 2.
Structure of the paper. In Section 1.1, for completeness of the theory, we discuss the p-Laplacian on Euclidean domains and Riemannian manifolds, and in Section 1.2 we recall the basic notions on oriented hypergraphs. In Section 2 we define the p-Laplacians for p > 1 and we establish their generalized min-max principle, and similarly, in Section 3, we introduce and discuss the 1-Laplacians for oriented hypergraphs. Furthermore, in Section 4 we discuss the smallest and largest eigenvalues of the p-Laplacians for all p, in Section 5 we prove two nodal domain theorems, and in Section 6 we discuss the smallest nonzero eigenvalue. Finally, in Section 7 we discuss several vertex partition problems and their relations to the p-Laplacian eigenvalues, while in Section 8 we discuss hyperedge partition problems.
In [23] we shall build upon the results developed in this paper.
Related work. It is worth mentioning that, in [18], other vertex p-Laplacians for hypergraphs have been introduced and studied. While these generalized vertex p-Laplacians coincide with the ones that we introduce here in the case of graphs, they do not coincide for general hypergraphs. Also, [18] focuses on classical hypergraphs, while we consider, more generally, oriented hypergraphs.

The p-Laplacian on Euclidean domains and Riemannian manifolds
There is a strong analogy between Laplace operators on Euclidean domains and Riemannian manifolds on one hand and their discrete versions on graphs and hypergraphs, and this is also some motivation for our work. Therefore, it may be useful to briefly summarize the theory on Euclidean domains and Riemannian manifolds.
Let Ω ⊂ R n be a bounded domain, with piecewise Lipschitz boundary ∂Ω, in order to avoid technical issues that are irrelevant for our purposes. More generally, Ω could also be such a domain in a Riemannian manifold.
Let first 1 < p < ∞. For u in the Sobolev space W 1,p (Ω), we may consider the functional Its Euler-Lagrange operator is the p-Laplacian for p = 2, we have, of course, the standard Laplace operator. Note that we use the − sign in (2) both to make the operator a positive one and to conform to the conventions used in this paper. The eigenvalue problem arises when we look for critical points of I p under the constraint or equivalently, if we seek critical points of the Rayleigh quotient among functions u ≡ / 0. To make the problem well formulated, we need to impose a boundary condition, and we consider here the Dirichlet condition u ≡ 0 on ∂Ω.
On a compact Riemannian manifold M with boundary ∂M , we can do the same when we integrate in (1), (3) with respect to the Riemannian volume measure, and let ∇ and div denote the Riemannian gradient and divergence operators. When ∂M = ∅, we do not need to impose a boundary condition.
Eigenfunctions and eigenvalues then have to satisfy the equation For 1 < p < ∞, the functionals in (1) and (3) are strictly convex, and the spectral theory is similar to that for p = 2, that is, the case of the ordinary Laplacian, which is a well studied subject. (See for instance [41] for the situation on a Riemannian manifold.) For p = 1, however, the functionals are no longer strictly convex, and things get more complicated. (6) then formally becomes − div ∇u |∇u| = λ u |u| .
In (6) for p > 1, we may put the right hand side = 0 at points where u = 0, but this is no longer possible in (7). This eigenvalue problem has been studied by Kawohl, Schuricht and their students and collaborators, as well as by Chang, and we shall summarize their results. Some references are [7, 24-30, 34, 37]. One therefore formally replaces (7) by defining substitute z of ∇u |∇u| and a substitute s of u |u| , leading to − divz = λs (8) where s ∈ L ∞ (Ω) satisfies s(x) ∈ Sgn(u(x)) (9) with Sgn(t) := if t < 0, and the vector field z ∈ L ∞ (Ω, R n ) satisfies z ∞ = 1, divz ∈ L n (Ω), − Ω u divzdx = I(u) (10) where Again (11) needs some explanation. In fact, while for p > 1, the natural space to work in is W 1,p (Ω), for p = 1, it is no longer W 1,1 (Ω), but rather BV (Ω). This space (for a short introduction, see for instance [20]) consists of all functions L 1 (Ω) for which Note that when u ∈ C 1 (Ω), we have and thus, BV -functions permit such an integration by parts in a weak sense. More precisely, for a BV -function u, its distributional gradient is represented by a finite R n valued signed measure |Du|dx, and we can write Also, u ∈ BV (Ω) has a well-defined trace u ∂Ω ∈ L 1 (∂Ω), and (13) generalizes to where ν is the outer unit normal of ∂Ω. Importantly, BV -functions can be discontinuous along hypersurfaces. A Borel set E ⊂ Ω has finite perimeter if its characteristic function χ E satisfies For instance, if the boundary of E is a compact Lipschitz hypersurface, then the perimeter of E is simply the Hausdorff measure H n−1 (∂E). And if E ⊂ Ω, we have The problem with (8), however, is that in general it has too many solutions, as it becomes rather arbitrary on sets of positive measure where u vanishes, see [30]. The solutions that one is really interested in should be the critical points of a variational principle, with the vanishing of the weak slope of [11] as the appropriate criterion. Inner variations provide another necessary criterion [30]. Viscosity solutions provide another criterion which, however, is still not stringent enough [26]. The Cheeger constant of Ω then is defined as where |E| is the Lebesgue measure of E. A set realizing the infimum in (16) is called a Cheeger set, and every bounded Lipschitz domain Ω possesses at least one Cheeger set. For such a Cheeger set E ⊂ Ω, ∂E ∩ Ω is smooth except possibly for a singular set of Hausdorff dimension at most n − 8 and of constant mean curvature 1 n−1 h 1 (Ω) at all regular points. When Ω is not convex, its Cheeger set need not be unique.
In fact, h 1 (Ω) equals the first eigenvalue of the 1-Laplacian. More precisely, is the smallest λ = 0 for which there is a nontrivial solution u of (7), and such a u is of the form χ E for a Cheeger set, up to a multiplicative factor, of course. Also, if λ 1,p (Ω) denotes the smallest nonzero eigenvalue of (6), then lim We also have the lower bound generalizing the original Cheeger bound for p = 2. More generally, for any family of eigenvalues λ k,p (Ω) of (6), lim p→1 + λ k,p (Ω) is an eigenvalue of (7). The converse is not true, however; (7) may have more solutions than can be obtained as limits of solutions of (6).
The functional |Du| appears also in image denoising, in so-called TV models (where the acronym TV refers to the fact that |Du|(Ω) is the total variation of the measure |Du|dx) introduced in [36]. There, one wants to denoise a function f : Ω → R by smoothing it, and in the TV models, one wants to minimize a functional of the form Ω |u − f | is the so-called fidelity term that controls the deviation of the denoised version u from the given data f . µ > 0 is a parameter that balances the smoothness and the fidelity term. Formally, a minimizer u has to satisfy an equation of the form div Du which is similar to (7). It turns out, however, that when such a model is applied to actual data, the performance is not so good, and it has been found preferable to modify (20) to what is called a nonlocal model in image processing [16]. In [19], such a model was derived from geometric considerations, and this may also provide some insight into the relation with the discrete models considered in this paper, we now recall the construction of that reference.
Let Ω be a domain in R n or some more abstract space, and ω : Ω × Ω → R a nonnegative, symmetric function. ω(x, y) can be interpreted as some kind of edge weight between the points x, y for any pair (x, y) ∈ Ω × Ω. Here x, y can also stand for patches in the image, and in our setting, they could also be vertices in a graph (in which case the integrals below would become sums). We define the averagē ω : Ω → R of ω byω and assume thatω is positive almost everywhere. On a graph, while ω is an edge function,ω would be a vertex function,ω(x) being the degree of the vertex x with edge weights ω(x, y). We first useω(x) and ω(x, y) to define the L 2 -norms for functions u : Ω → R and vector fields p, that is, p : Ω × Ω → R, (p 1 , p 2 ) L 2 := Ω×Ω p 1 (x, y)p 2 (x, y)ω(x, y)dxdy and the corresponding norms |u| and |p|.
The discrete derivative of a function (an image) u : Ω → R is defined by Even though Du does not depend on ω, it is in some sense analogous to a gradient, as we shall see below. Its pointwise norm then is given The divergence of a vector field p : Ω × Ω → R is defined by Note that, in contrast to Du for a function u, the divergence of a vector field depends on the weight ω. For u : Ω → R and p : Ω × Ω → R, we then have the analog of (13).
With the vector field Du and the divergence operator div, we can define a Laplacian for functions which in the case of a graph is the Laplacian we have been using. The nonlocal TV (or BV) functional of [19] then is This leads to the nonlocal TV model = Ω ( Ω (u(y) − u(x)) 2 ω(x, y)dy) 1 2 ω(x)dx + µ Ω Ω |u(x) − f (x)|ω(x, y)dxdy.
(28) It should be of interest to explore such models on hypergraphs. That would offer the possibility to account not only for correlations between pairs, but also between selected larger sets of vertices, for instance three collinear ones. 38]). An oriented hypergraph is a pair Γ = (V, H) such that V is a finite set of vertices and H is a set such that every element h in H is a pair of disjoint elements (h in , h out ) (input and output) in P(V ) \ {∅}. The elements of H are called the oriented hyperedges. Changing the orientation of a hyperedge h means exchanging its input and output, leading to the pair (h out , h in ).

Basic notions on hypergraphs
With a little abuse of notation, we shall see h as h in ∪ h out . Definition 1.2 ( [33]). Given h ∈ H, we say that two vertices i and j are co-oriented in h if they belong to the same orientation sets of h; we say that they are anti-oriented in h if they belong to different orientation sets of h. Definition 1.3. Given i ∈ V , we say that two hyperedges h and h ′ contain i with the same ; we say that they contain i with opposite orientation From now on, we fix such an oriented hypergraph Γ = (V, H) on n vertices 1, . . . , n and m hyperedges h 1 , . . . , h m . We assume that there are no vertices of degree zero. We denote by C(V ) the space of functions f : V → R and we denote by C(H) the space of functions γ : H → R.
2 p-Laplacians for p > 1 We define its eigenvalue problem as We say that a nonzero function f and real number λ satisfying (29) are an eigenfunction and the corresponding eigenvalue for ∆ p .
Remark 2.2. Definition 2.1 generalizes both the graph p-Laplacian and the normalized Laplacian defined in [21] for hypergraphs, which corresponds to the case p = 2.
Remark 2.3. The p-Laplace operators for classical hypergraphs that were introduced in [18] coincide with the vertex p-Laplacians that we introduced here in the case of simple graphs, but not in the more general case of hypergraphs. In fact, the Laplacians in [18] are related to the Lovász extension, while the operators that we consider here are defined via the incidence matrix. Also, the corresponding functionals for the p-Laplacians in [18] are of the form and these are non-smooth in general, even for p > 1. In our case, the corresponding functionals are of the form and these are smooth for p > 1.
We define its eigenvalue problem as We say that a nonzero function γ and a real number λ satisfying (30) are an eigenfunction and the corresponding eigenvalue for ∆ H p .
Remark 2.5. For p = 2, Definition 2.4 coincides with the one in [21]. Also, as we shall see, while it is known that the nonzero eigenvalues of ∆ p and ∆ H p coincide for p = 2, this is no longer true for a general p.

Generalized min-max principle
For p = 2, the Courant-Fischer-Weyl min-max principle can be applied in order to have a characterizations of the eigenvalues of ∆ 2 and ∆ H 2 in terms of the Rayleigh Quotients of the functions f ∈ C(V ) and γ ∈ C(H), respectively, as shown in [21]. In this section we prove that, for p > 1, a generalized version of the min-max principle can be applied in order to know more about the eigenvalues of ∆ p and ∆ H p . Similar results are already known for graphs, as shown for instance in [40]. Before stating the main results of this section, we define the generalized Rayleigh Quotients for functions on the vertex set and for functions on the hyperedge set.
Proof. For p ∈ R >1 , RQ p is differentiable on R n \ 0. Also, where we have used the fact that Hence, Furthermore, if f ′ is an eigenfunction corresponding to any eigenvalue λ, This proves the claim for ∆ p . The case of ∆ H p is similar. We have that Therefore, This proves the first implication for ∆ H p . The inverse implication is analogous to the case of ∆ p .
is the smallest (resp. largest) eigenvalue of ∆ p , and f realizing (31) is a corresponding eigenfunction.
Analogously, min is the smallest (resp. largest) eigenvalue of ∆ H p , and γ realizing (32) is a corresponding eigenfunction.
Proof. By Fermat's theorem, if f = 0 minimizes or maximizes RQ p over R n \ 0, then ∇RQ p (f ) = 0. The claim for ∆ p then follows by Theorem 2.8, and the case of ∆ H p is analogous.
We now give a preliminary definition, before stating the generalized min-max principle.
Definition 2.10. For a centrally symmetric set S in R n , its Krasnoselskii Z 2 genus is defined as For each k ≥ 1, we let Gen k := {S ⊂ R n : S centrally symmetric with gen(S) ≥ k}.
Remark 2.11. From the above definition we get an inclusion chain Therefore, the Krasnoselskii Z 2 genus gives a graded index of the family of all centrally symmetric sets with center at 0 in R n , which generalizes the (linear) dimension of subspaces.
are eigenvalues of ∆ p . They satisfy The same holds for the constants that are eigenvalues of ∆ H p .
Proof. By Theorem 2.8, in order to prove the claim for ∆ p it suffices to show that λ k (∆ p ) defined in (33) is a critical value of RQ p . Let be the p-norm with weights given by the degrees, and let where R + S := {cg : g ∈ S, c > 0}. Therefore, it can be verified that From the Liusternik-Schnirelmann Theorem applied to the smooth function E p restricted to the smooth l p -sphere S p it follows that such a min-max quantity must be an eigenvalue of E p on S p . This proves the claim for ∆ p . The case of ∆ H p is similar, if we consider Remark 2.13. For the case of p = 2, a linear subspace X in R n with dim X = k satisfies gen(X) = k and by considering the sub-family This coincides with the Courant-Fischer-Weyl min-max principle. On the other hand, for p > 1, we only know that In particular, while for p = 2 we know that the n eigenvalues of ∆ p (resp. the m eigenvalues of ∆ H p appearing in Theorem 2.12) are all the eigenvalues of ∆ p (resp. ∆ H p ), we don't know whether ∆ p and ∆ H p have also more eigenvalues, for p = 2. This is still an open question also for the graph case. In other words, we don't know whether all eigenvalues of ∆ p and ∆ H p can be written in the min-max Rayleigh Quotient form.
We formulate this conjecture, because for the p-Laplacian on domains and manifolds as well as on graphs, it is an open problem whether all the eigenvalues of the p-Laplacian are of the min-max form (see [4,6,13] and [40]). Thus, as far as we know, Conjecture 1 is open in both the continuous and the discrete setting.
Throughout the paper, given p > 1 we shall denote by the eigenvalues of ∆ p and ∆ H p , respectively, which are described in Theorem 2.12. We shall call them the min-max eigenvalues. Note that, although we cannot say a priori whether these are all the eigenvalues of the p-Laplacians, in view of Corollary 2.9 we can always say that

1-Laplacians
In this section we generalize the well known 1-Laplacian for graphs [7,8,17] to the case of hypergraphs.
Definition 3.1. The 1-Laplacian is the set-valued operator such that, given f ∈ C(V ), where e 1 , . . . , e n is the orthonormal basis of R n and Analogously, the hyperedge 1-Laplacian for functions γ ∈ C(H) is Remark 3.2. The 1-Laplacian is the limit of the p-Laplacian with respect to the set-valued upper limit, i.e.
where B δ (f ) is the ball with radius δ and center f . In other words, ∆ 1 f is the set of limit points of On the other hand, for a general f ∈ C(V ), the limit may not exist. To some extent, the set-valued upper limit ensures the upper semi-continuity of the family of p-Laplacians, that is, the set-valued mapping [1, Definition 3.3. The eigenvalue problem of ∆ 1 is to find the eigenpair (λ, f ) such that or equivalently, in terms of Minkowski summation, In coordinate form it means that there exist Remark 3.4. A shorter coordinate form of the eigenvalue problem for the 1-Laplacian is Observe also that ( i∈h in f (i) − i∈hout f (i))z h = | i∈h in f (i) − i∈hout f (i)| and f (i)z i = |f (i)|, for all h ∈ H and for all i ∈ V .
The eigenvalue problem of ∆ H 1 can be defined in an analogous way. In particular, all results shown in this section for ∆ 1 also hold for ∆ H 1 . Without loss of generality, we only prove them for ∆ 1 .
Definition 3.5. For the generalized Rayleigh Quotient RQ 1 (cf. Definition 2.6), its Clarke derivative at f ∈ C(V ) is This is a compact convex set in C(V ).
Remark 3.6. Clarke introduced such a derivative for locally Lipschitz functions, in the field of nonsmooth optimization [9,10]. Clearly, RQ 1 is not smooth, but it is piecewise smooth (therefore locally Lipschitz) on R n \ 0. Hence, the Clarke derivative for RQ 1 is well defined. Also, since the Clarke derivative coincides with the usual derivative for smooth functions, we choose to denote it by ∇ also for locally Lipschitz functions.
Finally, applying the additivity of Clarke's derivative, we derive the desired identities.
Theorem 3.9 (Min-max principle for the 1-Laplacian). If f is a critical point of the function RQ 1 , i.e. 0 ∈ ∇RQ 1 (f ), then f is an eigenfunction and RQ 1 (f ) is the corresponding eigenvalue of ∆ 1 . A function f ∈ C(V ) \ 0 is a maximum (resp. minimum) eigenfunction of ∆ 1 if and only if it is a maximizer (resp. minimizer) of RQ 1 ; λ is the largest (resp. smallest) eigenvalue of ∆ 1 if and only if it is the maximum (resp. minimum) value of RQ 1 . Also, the constants λ k (∆ 1 ) := inf are eigenvalues of ∆ 1 . Furthermore, lim p→1 + λ k (∆ p ) = λ k (∆ 1 ), and any limit point of {f k,p } p>1 is an for some k, l ∈ N + , then λ k (∆ 1 ) has the multiplicity at least l + 1.
Proof. The proof is based on the theory of Clarke derivative, established in [10].
Let f be a critical point of the function RQ 1 . By the chain rule for the Clarke derivative, Therefore, f is an eigenfunction of ∆ 1 , and RQ 1 (f ) is the corresponding eigenvalue. Also, again by the basic results on Clarke derivative, if f is a maximizer (minimizer) of RQ 1 , then 0 ∈ ∇RQ 1 (f ). Hence, 0 ∈ ∆ 1 f − RQ 1 (f )Sgn(f ). Thus, f is an eigenfunction, and RQ 1 (f ) is a corresponding eigenvalue.
The min-max principle (37) is a consequence of the nonsmooth version of the Liusternik-Schnirelmann Theorem [11], and thus we omit the details of the proof.
The convergence property lim p→1 + λ k (∆ p ) = λ k (∆ 1 ) is a consequence of the result on Gammaconvergence of minimax values [12]. Now, without loss of generality, we may assume that f k,p → f * , p → 1 + . Then, according to Remark 3.2, lim which means that f * is an eigenfunction of ∆ 1 .
The condition lim derives that λ k (∆ 1 ) has the multiplicity at least (l + 1) according to the Liusternik-Schnirelmann Theory. This completes the proof.
Analogously to the case of p > 1, also for p = 1 we shall denote by the eigenvalues of ∆ 1 that are described in Theorem 2.12 and the analogous eigenvalues of ∆ H 1 that can be obtained in the same way. Also in this case, as well as for p > 1, we can always say that Remark 3.10. In contrast to the case of the p-Laplacian for p > 1, the converse of Theorem 3.9 is not true, that is, there exist eigenfunctions f of ∆ 1 that are not a critical points of RQ 1 . However, showing this requires a long argument that we bring forward in [23]. In [23] we also show, furthermore, that Conjecture 1 cannot hold for ∆ 1 . (We had already noted in Section 1.1 that this is also a subtle issue in the continuous case.)

Smallest and largest eigenvalues
In [31], it has been proved that max γ∈C(H) Hence, we can characterize the maximal eigenvalue of ∆ H 1 in virtue of a combinatorial quantity. In this section we investigate further properties of both the largest and the smallest eigenvalues of the p-Laplacians, for general p. Proof. Letf : V → R that is 1 on a fixed vertex and 0 on all other vertices. Then, for all p, RQ p (f ) = 1. Therefore, The inverse inequality follows by Lemma 4.1.
Proof. letγ : H → R that is 1 on a fixed hyperedge h and 0 on all other hyperedges. Then, for all p, Therefore, Since this is true for all h, this proves the claim.

Nodal domain theorems
In [33], the authors prove two nodal domain theorems for ∆ 2 . In this section we establish similar results for ∆ p , for all p ≥ 1. Before, we recall the definitions of nodal domains for oriented hypergraphs. We refer the reader to [5] for nodal domain theorems on graphs.
A negative nodal domain of f is a connected component of H ∩ supp − (f ).

Signless nodal domain
Definition 5.2. We say an eigenvalue λ of ∆ p has multiplicity r if gen{eigenfunctions w.r.t. λ} = r.
Theorem 5.3. If f is an eigenfunction of the k-th min-max eigenvalue λ k (∆ p ) and this has multiplicity r, then the number of nodal domains of f is smaller than or equal to k + r − 1.
Proof. Suppose the contrary, that is, f is an eigenfunction of λ k with multiplicity r, and f has at least k + r nodal domains which are denoted by V 1 , . . . , V k+r . For simplicity, we assume that

Consider a linear function-space
Since V 1 , . . . , V k+r are pairwise disjoint, dim X = k+r. Given g ∈ X \0, there exists (t 1 , . . . , t k+r ) = 0 such that By the definition of nodal domain, each hyperedge h intersects with at most one V i ∈ {V 1 , . . . , V k+r }, which implies that Finally, we note that for p > 1, For the case of p = 1, we have in which the parameters z h ∈ Sgn( i∈h in f (i) − j∈hout f (j)) and z i ∈ Sgn(f (i)) (cf. Remark 3.4). Therefore, By the min-max principle for ∆ p , which leads to a contradiction. Proof. Suppose the contrary, that is, f is an eigenfunction of λ k with multiplicity r, and f has at least n − k + r + 1 nodal domains which are denoted by V 1 , . . . , V n−k+r+1 . Consider a linear function-space

Positive and negative nodal domain theorem
Since V 1 , . . . , V n−k+r+1 are pairwise disjoint, dim X = n − k + r + 1. For g ∈ X \ 0, there exists (t 1 , . . . , t n−k+r+1 ) = 0 such that g = n−k+r+1 i=1 t i f | V i . By definition of positive and negative nodal domains, each hyperedge h intersects at most one positive nodal domain and at most one negative nodal domain. Thus, for l = l ′ and h ∈ H, Now, with a little abuse of notation we let h = h in . For p > 1, we have that h∈H i∈h where the inequality is deduced by taking A = i∈h f | V l (i) and B = i∈h f | V ′ l (i) in the following lemma. Similarly, for p = 1 we have h∈H i∈h where z h ∈ Sgn( i∈h f (i)) and z i ∈ Sgn(f (i)).
It is only left to prove Lemma 5.5.
Proof of Lemma 5.5. Without loss of generality, we may assume that A > 0 > B and A > B ′ := |B|.

Smallest nonzero eigenvalue
In this section we discuss the smallest nonzero eigenvalue λ min of ∆ p , for p ≥ 1, as a continuation of Sections 5 and 6 in [33], which are focused on the easier study of λ min for the 2-Laplacian. As in [33], we let I h : V → R and I i : H → R be defined by otherwise.
Then ∇RQ p (f ) = 0. Consider the function t → RQ p (f − tg f ). On the one hand, On the other hand, E p (f − tg f ) = E p (f ) and the function is a strictly convex function with minimum at t = 1. This implies that (42) is strictly decreasing and convex on (−1, 1), thus Hence, we get d dt t=0 RQ p (f − tg f ) > 0, which leads to a contradiction. This proves the case p > 1. Finally, we complete the proof of the case p = 1. Since we only need to prove that (III) holds also for ∆ 1 . Suppose the contrary and letf be an eigenfunction corresponding to an eigenvalue λ ∈ (0,λ). Then, 0 ∈ ∇E 1 (f ) − λ∇ f 1 . Now, consider a flow near f defined by η(f, t) := f − tg f , where t ≥ 0 and f ∈ B δ (f ) for sufficiently small δ > 0. Note that is an increasing function of t, since f − tg f 1 < f 1 and · 1 is convex. Consequently, by the theory of weak slope [11], we have that 0 ∈ ∇(E 1 (f ) − λ f 1 ) = ∇E 1 (f ) − λ∇ f 1 , which is a contradiction. This completes the proof.
We shall now discuss some consequences of Theorem 6.1.
Proof. It follows immediately from Theorem 6.1.
Corollary 6.4. For p ≥ 1, let λ p,min be the smallest positive eigenvalue of the p-Laplacian. Then, Thus, applying Corollary 6.3, we have The case of p ≥ 2 is similar. Remark 6.5. We further haveλ p,min whereλ p,min = λ 1 p p,min . This implies that thusλ p,min is a continuous function of p ∈ [1, ∞) and the limit lim p→+∞λ p,min ∈ [0, n] exists.
Remark 6.6. For p ≥ 1, let By Corollary 6.3 and Remark 6.5, we get that which can be seen as a dual inequality with respect to the one in Corollary 6.3. Note that the constant C p is such that C 2 = 1 for all oriented hypergraphs and C 1 = 2 in the graph case.

Vertex partition problems
In [33], two vertex partition problems for oriented hypergraphs have been discussed: the k-coloring, that is, a function f : V → {1, . . . , k} such that f (i) = f (j) for all i = j ∈ h and for all h ∈ H, and the generalized Cheeger problem. In this section we discuss more partition problems and we also define a new coloring number that takes signs into account as well.
In [33], the generalized Cheeger constant is defined as

k-cut problems
We now generalize the balanced minimum k-cut problem and the max k-cut problem, known for graphs [15,35], to the case of hypergraphs.
Definition 7.2. Given k ∈ {2, . . . , n}, the balanced minimum k-cut is The maximum k-cut is Lemma 7.3. For each ∅ = S ⊆ V and for each p ≥ 1, Therefore, in particular, for each k ∈ {2, . . . , n} and Proof. Let f ∈ C(V ) be 1 on S and 0 onS. Then, The second claim follows by applying the first one to all the V i 's.

Signed coloring number
We now introduce the new notion of signed coloring number, that takes into account also the input/output structure of the hypergraph. We denote by χ(Γ) the coloring number defined in [33]. Corollary 7.6. Let χ sign := χ sign (Γ) and let V 1 , . . . , V χ sign be the corresponding coloring classes. For each p ≥ 1, Also, the upper bound in (43) shrinks to an equality for p = 1.
Proof. The first fact follows from Lemma 7.3 since, by definition of signed coloring number, for each coloring class V i .
In the particular case of Since we know, from Lemma 4.2, that max f RQ 1 (f ) = 1, this proves that the upper bound in (43) shrinks to an equality for p = 1.
Remark 7.7. The fact that the upper bound in (43) shrinks to an equality for p = 1 is particularly interesting because this is similar to what happens for the Cheeger constant h in the case of graphs, and for the Cheeger-like constant Q defined in [22] for graphs and generalized in [31] for hypergraphs.
In fact, we have that: 1. For connected graphs, the Cheeger constant h can be used for bounding λ 2 in the case of ∆ 2 and, as shown in [8,17], it is equal to λ 2 in the case of ∆ 1 .
2. For general hypergraphs, the Cheeger-like constant Q can be used for bounding λ n in the case of ∆ 2 and ∆ H 2 , and it is equal to λ n in the case of ∆ H 1 (cf. [31]).
3. In (43) we again have something similar, because the quantity that bounds λ n from below for ∆ p equals λ n for ∆ 1 .
Of course, the main difference between the last case and the first two is that h and Q are constants that are independent of p, while the quantity in (43) changes when p changes.

Multiway partitioning
In this section we generalize the notion of k-cut and we use it for bounding the smallest and largest eigenvalue of the classical Laplacian ∆ 2 .
Theorem 7.11. Let λ 1 and λ n be the smallest and the largest eigenvalue of the classical normalized Laplacian ∆ 2 , respectively. For any (k, l)-family, Proof. We first focus on the case that (S 1 , . . . , S k ) is a (k, l)-cover. For r ∈ {1, . . . , k}, define a function f r : V → R by where we have used the equality since each vertex in h is covered l times by S 1 , . . . , S k (l ≤ k).
We can verify that the minimum and maximum of the above quantity belong to k r=1 e(S r ) Vol(V ) .
To see this, we make the following observations.
Interestingly, applying Theorem 7.11 to a (k, 1)-cover of a graph, we get which relates to the max k-cut problem.

General partitions
Lemma 7.14. We have and .
Next, we give a lower bound for (46). By the convexity of t → |t| p , we have which implies |B − A| p ≥ c p−1 |B| p − ( c 1−c ) p−1 |A| p . Thus, Finally, the same method gives Corollary 7.15. The following constants are smaller than or equal to λ n (∆ p ): Proof. Taking t = k − 1 and c = 1 2 in (44), we have the first. Taking t = k − 1 and c = 1 k in (44), we get the middle one. Taking t = 1 and c = 1 2 in (44), we obtain the last one.
Corollary 7.16. The following constants are larger than or equal to λ 1 (∆ p ): Proof. Taking t = −1 in (45), we get the first constant. Letting t → ∞ in (45), we obtain the second one.

Hyperedge partition problems
While in the previous section we have discussed vertex partition problems and their relation to ∆ p , here we introduce the analogous hyperedge partition problems and their relations with ∆ H p . We start by defining, for each ∅ =Ĥ ⊂ H, a quantity e p (Ĥ) analogous to the quantity e p (S) defined for subsets of vertices. Namely, we let where, given i ∈ V , we let i in := #{ hyperedges in which i is an input }, i out := #{ hyperedges in which i is an output }.
Remark 8.1. Analogously to the vertex case, we can say that computing e p (Ĥ) means deleting all hyperedges in H \Ĥ and then computing e p on the hyperedge set of the sub-hypergraph obtained. It is therefore interesting to observe that, when e p is computed on H, where the first inequality is an equality if and only if each vertex is as often an input as an output, while the second one is an equality if and only if all vertices have the same sign for all hyperedges in which the graph is contained.
Furthermore, if the sub-hypergraphΓ := (V,Ĥ) of Γ is bipartite, without loss of generality we can assume that each vertex is either always an input or always an output for each hyperedge in which it is contained. In this case, Proof. By definition of signed hyperedge coloring number, for each coloring class H j . Together with Lemma 8.4, this proves the claim.