2026/01/14 by Kauê Cardoso
#math.CO
A line multigraph is obtained from a hypergraph by taking its hyperedges as vertices and joining two of them by as many edges as the number of vertices they share. We develop a matrix theory for line multigraphs of general, not necessarily uniform, hypergraphs. The central tool is the identity BTB = C + AL, where B is the incidence matrix, C is the diagonal matrix of hyperedge cardinalities and AL is the adjacency matrix of the line multigtaph. From this identity, we prove that the eigenvalues of the line multigraph of a hypergraph of rank r are at least -r, and we describe the eigenspace and the multiplicity of -r through an essential core of the hypergraph. We also give an explicit combinatorial condition under which -r is attained. As applications, we bound the spectral radius of the signless Laplacian matrix, characterizing the cases of equality, and we determine the complete signless Laplacian spectrum of a general power hypergraph. On the structural side, we show that connectivity, linearity, and regularity transfer between a hypergraph and its line multigraph, that every hypergraph shares its line multigraph with infinitely many others, and that each class of such hypergraphs contains a reduced representative.