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We say that a digraph is essentially cyclic if its Laplacian spectrum is notcompletely real. The essential cyclicity implies the presence of directedcycles, but not vice versa. The problem of characterizing essential cyclicityin terms of graph topology is difficult and yet unsolved. Its solution isimportant for some applications of graph theory, including that indecentralized control. In the present paper, this problem is solved withrespect to the class of digraphs with ring structure, which models some typicalcommunication networks. It is shown that the digraphs in this class areessentially cyclic, except for certain specified digraphs. The main technicaltool we employ is the Chebyshev polynomials of the second kind. A by-product ofthis study is a theorem on the zeros of polynomials that differ by one from theproducts of Chebyshev polynomials of the second kind. We also consider theproblem of essential cyclicity for weighted digraphs and enumerate the spanningtrees in some digraphs with ring structure.

Which Digraphs with Ring Structure are Essentially Cyclic?