A characterization of oriented hypergraphic balance via signed weak walks

An oriented hypergraph is a hypergraph where each vertex-edge incidence is given a label of +1 or −1, and each adjacency is signed the negative of the product of the incidences. An oriented hypergraph is balanced if the product of the adjacencies in each circle is positive.We provide a combinatorial interpretation for entries of kth power of the oriented hypergraphic Laplacian via the number of signed weak walks of length k. Using closed weak walks we prove a new characterization of balance for oriented hypergraphs and matrices that generalizes Harary's Theorem for signed graphs.