key: cord-0315003-yipwlyzo authors: Mulas, Raffaella title: Sharp bounds for the largest eigenvalue date: 2020-04-05 journal: nan DOI: 10.1134/s0001434621010120 sha: 07f6378846c6220335167d40166ff84f5ea550e9 doc_id: 315003 cord_uid: yipwlyzo We generalize the classical sharp bounds for the largest eigenvalue of the normalized Laplace operator, $frac{N}{N-1}leq lambda_Nleq 2$, to the case of chemical hypergraphs. 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. 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 [4] for the largest eigenvalue of graphs and prove that the lower bound Q ≤ λ N still holds also for hypergraphs. Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany. 1 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. Note that both definitions coincide with the usual notion of degree when we restrict to the graph case. We assume that deg v > 0 for each v ∈ V . Definition 2.4 ([1]). The normalized Laplacian associated to Γ is the operator 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. Remark 2.5. Because of the definition of degree that we are adopting and by definition of L, it is clear that, if a vertex v ∈ V does not belong to a hyperedge h ∈ H, then the normalized Laplacian of Γ coincides with the normalized Laplacian defined for Γ ′ , a hypergraph that is given by Γ with the additional assumption that v belongs to h as a catalyst. Furthermore, since we are assuming that deg v > 0 for each v ∈ V , we are not considering vertices that are catalysts for all hyperedges in which they are contained (and such vertices would produce the eigenvalue 0, as shown in [1] ). Therefore, without loss of generality we can focus on hypergraphs that do not have catalysts. We formalize this in the following lemma. Lemma 2.6. Let Γ = (V, H) be a chemical hypergraph such that there is no vertex that is a catalyst for all hyperedges in which it is contained. LetΓ := (V,Ĥ), wherê Then, Γ andΓ are isospectral. Proof. Since the degree does not take into account the hyperedges for which a vertex is a catalyst, it is clear by definition of L that Lf (v) is invariant in Γ and inΓ, for all v ∈ V and for all f : V → R. Therefore, in particular, the spectrum of L coincides for these two hypergraphs. In view of Lemma 2.6, without loss of generality we can focus on oriented hypergraphs, that is, chemical hypergraphs that do not include catalysts. Oriented hypergraphs have been introduced in [5] 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 [6, 7, 8, 9, 10, 11, 12, 13] . Throughout this paper we therefore work with a fixed connected oriented hypergraph (there are no catalysts) Γ = (V, H) on N nodes and M hyperedges. 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) . Definition 2.9. 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 We prove Theorem 3.1 in Section 4. Before, we discuss some consequences and examples. Remark 3.3. Observe that, in the graph case, |h| = 2 for each edge. Hence, in this case, (5) tells us that λ N ≤ 2, with equality if and only if the graph is bipartite. (5) is therefore a generalization of the classical upper bound for λ N , to the case of hypergraphs. Also, given a graph Γ, fix a vertex v and letΓ be the bipartite sub-graph of Γ given by the edges that have v as endpoint. Then, by (6) , Hence, from (6), we can re-infer the fact that λ N ≥ N N −1 for graphs. Example 3.4. Let Γ = K N be the complete graph on N nodes. Fix a vertex v and let Γ be the bipartite sub-graph of Γ given by the edges that have v as endpoint. Then, Therefore, (6) is an equality for K N . 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, with equality if and only if |f | is constant on all v ∈ h. 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 [4] 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 [4] , 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) can be generalized to the case of hypergraphs. In particular, the proof of Lemma 5.2 below generalizes the proof of [4, Lemma 4] . This proves the claim. Hypergraph Laplace operators for chemical reaction networks Spectral graph theory Coupled dynamics on hypergraphs: Master stability of steady states and synchronization Cheeger-like inequalities for the largest eigenvalue of the graph Laplace Operator An oriented hypergraphic approach to algebraic graph theory Oriented hypergraphs: Introduction and balance Spectral properties of oriented hypergraphs Intersection graphs of oriented hypergraphs and their matrices Oriented hypergraphic matrix-tree type theorems and bidirected minors via Boolean order ideals A characterization of oriented hypergraphic Laplacian and adjacency matrix coefficients Spectra of cycle and path families of oriented hypergraphs Lower bounds for the Laplacian spectral radius of an oriented hypergraph Incidence hypergraphs: Injectivity, uniformity, and matrix-tree theorems Extremal graph on normalized laplacian spectral radius and energy Bipartite and neighborhood graphs and the spectrum of the normalized graph Laplacian The dual Cheeger constant and spectra of infinite graphs Acknowledgments. The author is grateful to Aida Abiad and Jürgen Jost for the helpful comments, and to Alessandro Marfoni for giving her a roof during the COVID-19 outbreak. The results presented in this paper have been proved under that roof.