Spectral criterion for cycle balance in networks

A network is cycle balanced if the product of the weights (nonzero real numbers) of the lines of every cycle in it is positive. In this paper, we prove that a network D is cycle balanced if and only if its adjacency matrix is isospectral with its nonnegative counterpart. Consequent to this theorem is an analogous criterion for structural balance in sigraphs (abbreviation for “signed graphs”) as also for cycle balance in signed digraphs. These criteria establish in a natural way a wide scope for cospectrality considerations in the classes of signed digraphs and sigraphs.