Laplacians on quantum hypergraphs

Abstract

We introduce quantum hypergraphs, in analogy with the theory of quantum graphs developed over the last 15 years by many authors. We emphasize some problems that arise when one tries to define a Laplacian on a hypergraph. c 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim A hypergraph is a pair H := (V, E), where V is a set of nodes but, unlike in the more usual case of a graph, E is some set each of whose elements e -so-called hyperedges -is an unordered n e -tuple of elements of V, where each n e ∈ N may depend on e. Thus we obtain graphs as a special case of hypergraphs if n e ≡ 2. Conversely, a graph is canonically associated with H upon replacing each hyperedge e by a collection of 1 2 n e (n e − 1) edges that connect any two distinct elements of e: This graph is called the section of H in [1]. Observe that the connectivity of a hypergraph cannot be reconstructed from its section.If an orientation is assigned to each edge of a simple graph G := (V, E), i.e., if each edge is regarded as an ordered pair e ≡ (e init , e term ) ∈ V × V \ {(v, v) : v ∈ V}, then it is possible to define the incidence matrix I = (ι ve ) byotherwise, and the graph Laplacian by L := II T . Of course, L is a symmetric and positive semidefinite operator on the node space R V . The first attempts to develop a systematic theory of operators on R V -and in particular of graph Laplacians -date back to the 1970s: This spectral graph theory is nowadays rather mature, cf. [3,8]. One may want to introduce a hypergraph Laplacian as the graph Laplacian of the non-oriented hypergraph's section. This choice is made e.g. in [16] and a few subsequent papers on spectral clustering, but we argue that in this way the essential non-binary structure of a hypergraph is lost.An alternative approach seems to be more accurate and ultimately more appropriate for our purposes: Oriented hypergraphs have been introduced independently by many authors, see e.g. the historical remarks in [7], but it was only in the last few years that an algebraic hypergraph theory has been developed by their means. Indeed, upon regarding each hyperedge as an oriented pair e ≡ (e init , e term ) of disjoint subsets of V it is possible to introduce an incidence matrix I = (ι ve ) byotherwise.The algebraic properties of a hypergraph's incidence matrix are not as well understood as in the graph case. E.g., the combinatorial meaning of the rank and the co-rank of I seems to be unknown. Nevertheless, it is again possible to define a symmetric, positive semidefinite hypergraph Laplacian by L := II T . Its algebraic properties have been studied in [6] and more recently in [9,13,14], while here we briefly dip into the time-continuous evolution equationThe associated Cauchy problem is well-posed if the hypergraph is finite, but in general it is not associated with a Markov process, as one sees already in the simple case of a hypergraph consisting of three nodes and one hyperedge e with e int = {v 1 , v 2 } and e term = {v 3 }. In this case the incidence matrix, the hypergraph Laplacian, and t...