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Analysis on Graphs and Its Applications
Edited by: Pavel Exner, Academy of Sciences, Rez near Prague, Czech Republic, Jonathan P. Keating, University of Bristol, Clifton, Bristol, United Kingdom, Peter Kuchment, Texas A & M University, College Station, TX, Toshikazu Sunada, Meiji University, Kawasaki, Japan, and Alexander Teplyaev, University of Connecticut, Storrs, CT

Proceedings of Symposia in Pure Mathematics
2008; 705 pp; hardcover
Volume: 77
ISBN-10: 0-8218-4471-7
ISBN-13: 978-0-8218-4471-7
List Price: US$139
Member Price: US$111.20
Order Code: PSPUM/77
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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 nano-structures, metal-insulator 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, zeta-functions 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.


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 graphs--How 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 self-similar 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 Sturm-Liouville operators with singular potentials on star-type 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
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 semi-conducting polymers
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