Proceedings of Symposia in Pure Mathematics 2008; 705 pp; hardcover Volume: 77 ISBN10: 0821844717 ISBN13: 9780821844717 List Price: US$139 Member Price: US$111.20 Order Code: PSPUM/77
 This book addresses a new interdisciplinary area emerging on the border between various areas of mathematics, physics, chemistry, nanotechnology, and computer science. The focus here is on problems and techniques related to graphs, quantum graphs, and fractals that parallel those from differential equations, differential geometry, or geometric analysis. Also included are such diverse topics as number theory, geometric group theory, waveguide theory, quantum chaos, quantum wire systems, carbon nanostructures, metalinsulator transition, computer vision, and communication networks. This volume contains a unique collection of expert reviews on the main directions in analysis on graphs (e.g., on discrete geometric analysis, zetafunctions on graphs, recently emerging connections between the geometric group theory and fractals, quantum graphs, quantum chaos on graphs, modeling waveguide systems and modeling quantum graph systems with waveguides, control theory on graphs), as well as research articles. Readership Graduate students and research mathematicians interested in various areas of analysis, dynamical systems, groups and their actions on graphs, mathematical physics. Table of Contents Analysis on combinatorial graphs Review articles  R. Band, I. Oren, and U. Smilansky  Nodal domains on graphsHow to count them and why?
 M. D. Horton, H. M. Stark, and A. A. Terras  Zeta functions of weighted graphs and covering graphs
 T. Sunada  Discrete geometric analysis
Research articles  M. J. Gruber, D. H. Lenz, and I. Veselić  Uniform existence of the integrated density of states for combinatorial and metric graphs over \(Z^d\)
 D. Guido, T. Isola, and M. L. Lapidus  Bartholdi zeta functions for periodic simple graphs
 M. Kelbert and Y. Suhov  Asymptotic properties of Markov processes on Cayley trees
Analysis on fractals Review articles  V. Nekrashevych and A. Teplyaev  Groups and analysis on fractals
Research articles  R. Grigorchuk and Z. Šunić  Schreier spectrum of the Hanoi Towers group on three pegs
 A. Grigor'yan and T. Kumagai  On the dichotomy in the heat kernel two sided estimates
 M. L. Lapidus and E. P. J. Pearse  Tube formulas for selfsimilar fractals
 R. Peirone  Existence of Eigenforms on nicely separated fractals
Analysis on quantum graphs Review articles  J. Bolte and S. Endres  Trace formulae for quantum graphs
 J. Harrison  Quantum graphs with spin Hamiltonians
 J. P. Keating  Quantum graphs and quantum chaos
 P. Kuchment  Quantum graphs: An introduction and a brief survey
Research articles  G. Berkolaiko  Two constructions of quantum graphs and two types of spectral statistics
 B. M. Brown, M. S. P. Eastham, and I. G. Wood  An example on the discrete spectrum of a star graph
 B. M. Brown, M. Langer, and K. M. Schmidt  The HELP inequality on trees
 R. Carlson  Boundary value problems for infinite metric graphs
 T. Ekholm, R. L. Frank, and H. Kovařík  Remarks about Hardy inequalities on metric trees
 H. Flechsig and S. Gnutzmann  On the spectral gap in Andreev graphs
 G. Freiling, M. Ignatiev, and V. Yurko  An inverse spectral problem for SturmLiouville operators with singular potentials on startype graphs
 M. J. Gruber, M. Helm, and I. Veselić  Optimal Wegner estimates for random Schrödinger operators on metric graphs
 V. Kostrykin, J. Potthoff, and R. Schrader  Contraction semigroups on metric graphs
 K. Pankrashkin  Localization in a quasiperiodic model on quantum graphs
 O. Post  Equilateral quantum graphs and boundary triples
 B. Winn  A conditionally convergent trace formula for quantum graphs
Applications Review articles  S. Avdonin  Control problems on quantum graphs
 P. Exner  Leaky quantum graphs: A review
 D. Grieser  Thin tubes in mathematical physics, global analysis and spectral geometry
 O. Hul, M. Ławniczak, S. Bauch, and L. Sirko  Simulation of quantum graphs by microwave networks
 D. Krejčiřík  Twisting versus bending in quantum waveguides
Research articles  B. Bellazzini, M. Burrello, M. Mintchev, and P. Sorba  Quantum field theory on star graphs
 H. D. Cornean, P. Duclos, and B. Ricaud  On the skeleton method and an application to a quantum scissor
 S. A. Fulling and J. H. Wilson  Vacuum energy and closed orbits in quantum graphs
 P. Schapotschnikow and S. Gnutzmann  Spectra of graphs and semiconducting polymers
