12 research outputs found
Spectral Properties of Oriented Hypergraphs
An oriented hypergraph is a hypergraph where each vertex-edge incidence is
given a label of or . The adjacency and Laplacian eigenvalues of an
oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and
Laplacian matrices of an oriented hypergraph which depend on structural
parameters of the oriented hypergraph are found. An oriented hypergraph and its
incidence dual are shown to have the same nonzero Laplacian eigenvalues. A
family of oriented hypergraphs with uniformally labeled incidences is also
studied. This family provides a hypergraphic generalization of the signless
Laplacian of a graph and also suggests a natural way to define the adjacency
and Laplacian matrices of a hypergraph. Some results presented generalize both
graph and signed graph results to a hypergraphic setting.Comment: For the published version of the article see
http://repository.uwyo.edu/ela/vol27/iss1/24
Spectral Properties of Complex Unit Gain Graphs
A complex unit gain graph is a graph where each orientation of an edge is
given a complex unit, which is the inverse of the complex unit assigned to the
opposite orientation. We extend some fundamental concepts from spectral graph
theory to complex unit gain graphs. We define the adjacency, incidence and
Laplacian matrices, and study each of them. The main results of the paper are
eigenvalue bounds for the adjacency and Laplacian matrices.Comment: 13 pages, 1 figure, to appear in Linear Algebra App
Spectra of Complex Unit Hypergraphs
A complex unit hypergraph is a hypergraph where each vertex-edge incidence is
given a complex unit label. We define the adjacency, incidence, Kirchoff
Laplacian and normalized Laplacian of a complex unit hypergraph and study each
of them. Eigenvalue bounds for the adjacency, Kirchoff Laplacian and normalized
Laplacian are also found. Complex unit hypergraphs naturally generalize several
hypergraphic structures such as oriented hypergraphs, where vertex-edge
incidences are labelled as either or , as well as ordinary
hypergraphs. Complex unit hypergraphs also generalize their graphic analogues,
which are complex unit gain graphs, signed graphs, and ordinary graphs
Spectra of complex unit hypergraphs
A complex unit hypergraph is a hypergraph where each vertex-edge incidence is given a complex unit label. We define the adjacency, incidence, Kirchoff Laplacian and normalized Laplacian of a complex unit hypergraph and study each of them. Eigenvalue bounds for the adjacency, Kirchoff Laplacian and normalized Laplacian are also found. Complex unit hypergraphs naturally generalize several hypergraphic structures such as oriented hypergraphs, where vertex-edge incidences are labelled as either +1 or β1, as well as ordinary hypergraphs. Complex unit hypergraphs also generalize their graphic analogues, which are complex unit gain graphs, signed graphs, and ordinary graphs
An oriented hypergraphic approach to algebraic graph theory
AbstractAn oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of +1 or -1. We define the adjacency, incidence and Laplacian matrices of an oriented hypergraph and study each of them. We extend several matrix results known for graphs and signed graphs to oriented hypergraphs. New matrix results that are not direct generalizations are also presented. Finally, we study a new family of matrices that contains walk information