We consider the Dirac equation on periodic networks (quantum graphs). The self-adjoint quasi boundary conditions are derived. secular allowing us to find energy spectrum of particles quantum graphs is obtained. Band spectra different topologies calculated. Universality probability be in for certain graph observed.

We develop eigenvalue estimates for the Laplacians on discrete and metric graphs using different types of boundary conditions at the vertices of the metric graph. Via an explicit correspondence of the equilateral metric and discrete graph spectrum (also in the “exceptional” values of the metric graph corresponding to the Dirichlet spectrum) we carry over these estimates from the metric graph La...

The lexicographic product, a powerful binary operation in graph theory, offers methods for creating novel by establishing connections between each vertex of one and every another. Beyond its fundamental nature, this is found various applications across computer science disciplines, including network analysis, data mining, optimization. In paper, we give definition the weight function to product...

The Exterior-Interior duality expresses a deep connection between the Laplace spectrum in bounded and connected domains in R, and the scattering matrices in the exterior of the domains. Here, this link is extended to the study of the spectrum of the discrete Laplacian on finite graphs. For this purpose, two methods are devised for associating scattering matrices to the graphs. The Exterior -Int...

1 Laplacian Methods: An Overview 2 1.1 De…nition: The Laplacian operator of a Graph . . . . . . . . . . 2 1.2 Properties of the Laplacian and its Spectrum . . . . . . . . . . . 4 1.2.1 Spectrum of L and e L: Graph eigenvalues and eigenvectors: 4 1.2.2 Other interesting / useful properties of the normalized Laplacian (Chung): . . . . . . . . . . . . . . . . . . . . . 6 1.2.3 Laplacians of Weight...