For a given hypergraph, an orientation can be assigned to the vertex-edge incidences. This orientation is used to define the adjacency and Laplacian matrices. In addition to studying these matrices, several related structures are investigated includi
ng the incidence dual, the intersection graph (line graph), and the 2-section. A connection is then made between oriented hypergraphs and balanced incomplete block designs.
A theory of orientation on gain graphs (voltage graphs) is developed to generalize the notion of orientation on graphs and signed graphs. Using this orientation scheme, the line graph of a gain graph is studied. For a particular family of gain graphs
with complex units, matrix properties are established. As with graphs and signed graphs, there is a relationship between the incidence matrix of a complex unit gain graph and the adjacency matrix of the line graph.
An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of $+1$ or $-1$. The adjacency and Laplacian eigenvalues of an oriented hypergraph are studied. Eigenvalue bounds for both the adjacency and Laplacian matrices o
f 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.
An 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 kno
wn 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.
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 u
nit 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.
We obtain new bounds for the Laplacian spectral radius of a signed graph. Most of these new bounds have a dependence on edge sign, unlike previously known bounds, which only depend on the underlying structure of the graph. We then use some of these b
ounds to obtain new bounds for the Laplacian and signless Laplacian spectral radius of an unsigned graph by signing the edges all positive and all negative, respectively.
