Line Multigraphs of Hypergraphs

Abstract: A line multigraph is obtained from a hypergraph as follows: the vertices of the multigraph correspond to the hyperedges of the hypergraph, and the number of edges between two vertices is given by the number of vertices shared by the corresponding hyperedges. In this paper, we establish several structural and spectral properties of this class of multigraphs. More precisely, we show that important structural characteristics, such as connectivity, linearity, and regularity are, in some sense, preserved between a hypergraph and its line multigraph. We also prove that the eigenvalues of the line multigraph associated with a general hypergraph of rank $r$ are greater than or equal to $-r$, which generalizes a fundamental spectral property of line graphs. Furthermore, we provide sufficient conditions for $-r$ to be an eigenvalue of the line multigraph. Finally, we present applications of line multigraphs to the spectral theory of hypergraphs, including bounds for the signless Laplacian spectral radius of a hypergraph and a characterization of the signless Laplacian spectrum for a specific class of hypergraphs.