Investigation and application of Laplace spectra of orgraphs with the ring structure

The Laplace matrix is a square matrix $L=(\ell_{ij})\in\mathbb{R}^{n\times n}$ in which all nondiagonal elements are nonpositive and all row sums are equal to zero. Each Laplace matrix corresponds to a weighted orgraph with positive arc weights. The problem of reality of Laplace matrix spectrum for orgraphs of a special type consisting of two “counter” Hamiltonian cycles in one of which one or two arcs are removed is studied. Characteristic polynomials of Laplace matrices of these orgraphs are expressed through polynomials $Z_n(x)$ that can be obtained from Chebyshev second-kind polynomials $P_{2n}(y)$ by the substitution of $y^2=x$. The obtained results relate to properties of the product of Chebyshev second-kind polynomials. A direct method for computing the spectrum of Laplace circuit matrix is given. The obtained results can be used for computing the number of spanning trees in orgraphs of the studied type. One of the possible practical applications of these results is the investigation of topology and development of new Internet protocols.