In [1] , the author together with Jürgen Jost introduced the notion of chemical hypergraph, that is, a hypergraph with the additional structure that each vertex v in a hyperedge h is either an input, an output or both (in which case we say that v is a catalyst for h). They also defined, on such hypergraphs, a normalized Laplace operator that generalizes the one introduced by Chung for graphs [2] and they investigated some properties of its spectrum. Furthermore, in a recent work [3] , the author together with Christian Kuehn and Jürgen Jost proposed an application of this theory to the study of dynamical systems on hypergraphs.
Here we bring forward the study of the spectral properties of the hypergraph Laplacian. Particularly, we focus on the largest eigenvalue and we generalize the classical sharp bounds that are well known for graphs. As Chung showed in [2] , given a connected graph Γ on N nodes, its largest eigenvalue λ N is such that
with equality if and only if Γ is bipartite, and
with equality if and only if Γ is complete. Therefore, we can say that 2 − λ N estimates how different the graph is from being bipartite, while λ N − N N −1 quantifies how different it is from being complete. Here we generalize (1) and (2) to the case of hypergraphs. The results presented here hold, in particular, for oriented hypergraphs, that is, chemical hypergraphs that do not include catalysts. Oriented hypergraphs have been introduced in [4] by Reff and Rusnak, who also introduced the non-normalized Laplacian and the adjacency matrix for such hypergraphs. The spectral properties of these operators have been widely investigated, see for instance [5, 6, 7, 8, 9, 10, 11, 12] .
Structure of the paper. In Section 2 we recall some definitions from [1] and we fix some new notation and terminology. In Section 3 we state our main theorem and we prove it in Section 4. Finally, in Section 5, we discuss a corollary of our main theorem, namely, we can generalize the Cheeger-like constant Q introduced in [13] for the largest eigenvalue of graphs and prove that the lower bound Q ≤ λ N still holds also for hypergraphs.
Before stating our main results, we recall some basic definitions from [1] and we give a few new definitions that shall be useful for our discussion. •
We fix, from now on, a connected 1 (chemical) hypergraph Γ = (V, H) on N vertices and M hyperedges. We define the degree of a vertex v as deg v := hyperedges containing v only as an input or only as an output and we define the cardinality of a hyperedge h as |h| := vertices in h that are either only an input or only an output .
Note that, in [1] , the degree of a vertex is defined as the total number of hyperedges containing it (also as a catalyst). Here we consider this alternative definition of degree because it is more convenient in order to state our main results. However, the results about the spectrum of the Laplacian in [1] do not change if we consider this new definition. Furthermore, both definitions coincide with the usual notion of degree when we restrict to the graph case. We also assume that deg v > 0 for each v ∈ V , that is, there are no vertices that are catalysts for each hyperedge they belong to.
We recall that L has N real, non-negative eigenvalues that we denote by
These eigenvalues are invariant under changing the orientation of any hyperedge and, in particular, the largest eigenvalue on which we shall focus here can be written as
The functions f : V → R realizing (3) are the eigenfunctions of L for λ N . We say that the functions γ : H → R realizing (4) are the hyperedge-eigenfunctions. These are the eigenfunctions of the hyperedge-Laplacian, an operator that has the same nonzero spectrum of L and therefore the same largest eigenvalue. We refer the reader to [1] for more details.
Definition 2.5 ([1]). We say that a hypergraph Γ is bipartite if one can decompose the vertex set as a disjoint union V = V 1 ⊔ V 2 such that, for every hyperedge h of Γ, either h has all its inputs in V 1 and all its outputs in V 2 , or vice versa ( Figure 1) . 
for some vertices v i 's that are catalysts in h j . The index set I j might also be empty, in which caseĥ j = h j .
Definition 2.7. Given a sub-hypergraphΓ ⊂ Γ, we let
where degΓ(v) denotes the degree of v inΓ and |Ĥ| is the number of hyperedges inΓ.
We need the quantity η(Γ) defined above for the statement of Theorem 3.1 below.
We can now state our main theorem.
Theorem 3.1. For every hypergraph Γ,
with equality if and only if Γ is bipartite and |h| is constant for all h, and 
Therefore, (6) is an equality for K N .
Example 3.5. Let Γ = K N \ {(v 1 , v 2 )} be the complete graph with an edge (v 1 , v 2 ) removed. We know, from [14] , that λ N = N +1 N −1 . LetΓ be the bipartite sub-graph of Γ given by the edges that have either v 1 or v 2 as endpoint. Then,
Therefore, (6) is an equality also in this case.
Example 3.6. For a bipartite hypergraph Γ such that |h| = c is constant for each h, by Theorem 3.1 λ N = c. Also,
Therefore, (6) is an equality.
We split the proof of Theorem 3.1 into two steps: Lemma 4.1 and Lemma 4.2 below.
Proof. Given a bipartite sub-hypergraphΓ ⊂ Γ, let γ ′ : H → R be 1 onĤ and 0 otherwise. Then, up to changing (without loss of generality) the orientations of the hyperedges,
Since the above inequality is true for allΓ, this proves the claim. Proof. Let f : V → R be an eigenfunction for λ N . Then,
with equality if and only if f has its nonzero values on a bipartite sub-hypergraph. Now, for each h ∈ H, letĥ be h without catalysts. Then,
with equality if and only if |f | is constant on all v ∈ĥ. Therefore,
where the first inequality is an equality if and only if |f | is constant (since we assuming that Γ is connected), and the last inequality is an equality if and only if |h| is constant for all h. Putting everything together, we have that
with equality if and only if |h| is constant for all |h| while |f | is constant and it's defined on a bipartite sub-hypergraph (that is, |f | is constant and Γ is bipartite).
As a consequence of Theorem 3.1, we can also generalize the Cheeger-like constant introduced in [13] for the case of graphs,
where E is the edge set of the graph, and we can prove that the lower bound Q ≤ λ N still holds also for hypergraphs. Furthermore, we can also show that the characterization of Q proved in [13] ,
can be extended for hypergraphs as well. Note that (8) tells us that, for graphs, we can characterize Q by looking at the characterization of λ N in (3) and then replacing the L 2 -norm by the L 1 -norm both in the numerator and denominator. The reason why this is interesting is that something analogous happens to the classical graph Cheeger constant h. It is in fact well known that, for connected graphs, h bounds the first nonzero eigenvalue 2 λ 2 both above and below and it can be characterized by first looking at a characterization of λ 2 using the Rayleigh quotient and then replacing the L 2 -norm by the L 1 -norm both in the numerator and denominator [2] . Furthermore, the first Cheeger-like constant for the largest graph eigenvalue that has been introduced is the dual Cheeger constanth [15, 16] . What makes the two Cheeger-like constants conceptually different is the fact thath is related to the Cheeger-constant h [15] and it doesn't have a characterization analogous to the one of Q, in terms of the Rayleigh quotient.
In particular, for hypergraphs, we generalize (7) by defining
As a direct consequence of Theorem 3.1, we can prove the following corollary. 
We conclude by proving that also the characterization of Q in (8) This proves the claim.