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A hypergraph is said to be oriented if each edge-vertex incidence has a label of +1 or −1. An oriented hypergraph is called incidence balanced if there exists a bipartition of the vertex set such that every edge intersects one part of the bipartition in positively incident vertices with the edge and the other part in negatively incident vertices with the edge. In this paper, we investigate the incidence balance of oriented hypergraphs and induced signed hypergraphs by means of the eigenvalues of associated matrices and tensors and provide a spectral method to characterize the incidence balanced oriented hypergraphs and their induced signed hypergraphs. [ABSTRACT FROM AUTHOR]

Letbe a signed graph and letbe the adjacency matrix of. The nullity ofis the multiplicity of eigenvalue zero in the spectrum of. In this paper, we characterize the signed graphs of orderwith nullityor, and introduce a graph transformation which preserves the nullity. As an application we determine the unbalanced bicyclic signed graphs of orderwith nullityor, and signed bicyclic graphs (including simple bicyclic graphs) of orderwith nullity. [ABSTRACT FROM AUTHOR]

Using the result on Fiedler vectors of a simple graph, we obtain a property on the structure of the eigenvectors of a nonsingular unicyclic mixed graph corresponding to its least eigenvalue. With the property, we get some results on minimizing and maximizing the least eigenvalue over all nonsingular unicyclic mixed graphs on n vertices with fixed girth. In particular, the graphs which minimize and maximize, respectively, the least eigenvalue are given over all such graphs with girth 3. [ABSTRACT FROM AUTHOR]