On construction of upper and lower bounds for the HOMO-LUMO spectral gap
Soña Pavlíková Daniel Ševčovič
Numerical Algebra, Control & Optimization 2019, 9(1): 53-69 doi: 10.3934/naco.2019005

In this paper we study spectral properties of graphs which are constructed from two given invertible graphs by bridging them over a bipartite graph. We analyze the so-called HOMO-LUMO spectral gap which is the difference between the smallest positive and largest negative eigenvalue of the adjacency matrix of a graph. We investigate its dependence on the bridging bipartite graph and we construct a mixed integer semidefinite program for maximization of the HOMO-LUMO gap with respect to the bridging bipartite graph. We also derive upper and lower bounds for the optimal HOMO-LUMO spectral graph by means of semidefinite relaxation techniques. Several computational examples are also presented in this paper.

keywords: Invertible graph; bridged graph Schur complement mixed integer semidefinite programming spectral estimates HOMO-LUMO spectral gap
Area preserving geodesic curvature driven flow of closed curves on a surface
Miroslav KolÁŘ Michal BeneŠ Daniel ŠevČoviČ
Discrete & Continuous Dynamical Systems - B 2017, 22(10): 3671-3689 doi: 10.3934/dcdsb.2017148

We investigate a non-local geometric flow preserving surface area enclosed by a curve on a given surface evolved in the normal direction by the geodesic curvature and the external force. We show how such a flow of surface curves can be projected into a flow of planar curves with the non-local normal velocity. We prove that the surface area preserving flow decreases the length of the evolved surface curves. Local existence and continuation of classical smooth solutions to the governing system of partial differential equations is analysed as well. Furthermore, we propose a numerical method of flowing finite volume for spatial discretization in combination with the Runge-Kutta method for solving the resulting system. Several computational examples demonstrate variety of evolution of surface curves and the order of convergence.

keywords: Geodesic curvature driven flow surface area preserving flow Hölder smooth solutions flowing finite volume method

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