This volume is a collection of articles dedicated to quantum graphs, a newly emerging interdisciplinary field related to various areas of mathematics and physics. The reader can find a broad overview of the theory of quantum graphs. The articles present methods coming from different areas of mathematics: number theory, combinatorics, mathematical physics, differential equations, spectral theory, global analysis, and theory of fractals. They also address various important applications, such as Anderson localization, electrical networks, quantum chaos, mesoscopic physics, superconductivity, optics, and biological modeling.
Fluctuation-based proof of the stability of ac spectra of random operators on tree graphs by M. Aizenman, R. Sims, and S. Warzel Metrized graphs, Laplacian operators, and electrical networks by M. Baker and X. Faber Form factor expansion for large graphs: A diagrammatic approach by G. Berkolaiko The spectral form factor for quantum graphs with spin-orbit coupling by J. Bolte and J. Harrison Linear network models related to blood flow by R. Carlson Localization on Avron-Exner-last graphs: I. Local perturbations by K. Chen, S. Molchanov, and B. Vainberg Weighted Laplacians and the sigma function of a graph by F. Chung and R. M. Richardson Approximations of permutation-symmetric vertex couplings in quantum graphs by P. Exner and O. Turek Resistance of random Sierpinski gaskets by D. Fontaine, T. Smith, and A. Teplyaev Small diffusion asymptotics for exit problems on graphs by M. Freidlin and M. Weber Determinant of the Schrodinger operator on a metric graph by L. Friedlander Local spectral density and vacuum energy near a quantum graph vertex by S. A. Fulling What are zeta functions of graphs and what are they good for? by M. D. Horton, H. M. Stark, and A. A. Terras Fluctuation statistics for quantum star graphs by J. P. Keating Laplacians on metric graphs: Eigenvalues, resolvents and semigroups by V. Kostrykin and R. Schrader Transition from a network of thin fibers to the quantum graph: An explicitly solvable model by S. Molchanov and B. Vainberg On the limiting absorption principle and spectra of quantum graphs by B.-S. Ong Quantum mechanics, superconductivity and fluid flow in narrow networks by J. Rubinstein A relation between the bond scattering matrix and the spectral counting function for quantum graphs by H. Schanz The quantum graph as a limit of a network of physical wires by U. Smilansky and M. Solomyak On the trace formula for quantum star graphs by B. Winn.