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Hardcover ISBN:  9780821844717 
Product Code:  PSPUM/77 
List Price:  $148.00 
MAA Member Price:  $133.20 
AMS Member Price:  $118.40 
eBook ISBN:  9780821893845 
Product Code:  PSPUM/77.E 
List Price:  $139.00 
MAA Member Price:  $125.10 
AMS Member Price:  $111.20 
Hardcover ISBN:  9780821844717 
eBookISBN:  9780821893845 
Product Code:  PSPUM/77.B 
List Price:  $287.00$217.50 
MAA Member Price:  $258.30$195.75 
AMS Member Price:  $229.60$174.00 

Book DetailsProceedings of Symposia in Pure MathematicsVolume: 77; 2008; 705 ppMSC: Primary 05; 11; 20; 28; 35; 47; 58; 68; 70; 78; 81;
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.ReadershipGraduate students and research mathematicians interested in various areas of analysis, dynamical systems, groups and their actions on graphs, mathematical physics.

Table of Contents

Chapter 1. Analysis on combinatorial graphs

Review articles

Ram Band, Idan Oren and Uzy Smilansky  Nodal domains on graphs—how to count them and why? [ MR 2459862 ]

Matthew D. Horton, H. M. Stark and Audrey A. Terras  Zeta functions of weighted graphs and covering graphs [ MR 2459863 ]

Toshikazu Sunada  Discrete geometric analysis [ MR 2459864 ]

Research articles

Michael J. Gruber, Daniel H. Lenz and Ivan Veselić  Uniform existence of the integrated density of states for combinatorial and metric graphs over $\mathbb {Z}^d$ [ MR 2459865 ]

Daniele Guido, Tommaso Isola and Michel L. Lapidus  Bartholdi zeta functions for periodic simple graphs [ MR 2459866 ]

M. Kelbert and Y. Suhov  Asymptotic properties of Markov processes on Cayley trees [ MR 2459867 ]

Chapter 2. Analysis on fractals

Review articles

Volodymyr Nekrashevych and Alexander Teplyaev  Groups and analysis on fractals [ MR 2459868 ]

Research articles

Rostislav Grigorchuk and Zoran Šunić  Schreier spectrum of the Hanoi Towers group on three pegs [ MR 2459869 ]

Alexander Grigor′yan and Takashi Kumagai  On the dichotomy in the heat kernel two sided estimates [ MR 2459870 ]

Michel L. Lapidus and Erin P. J. Pearse  Tube formulas for selfsimilar fractals [ MR 2459871 ]

Roberto Peirone  Existence of eigenforms on nicely separated fractals [ MR 2459872 ]

Chapter 3. Analysis on quantum graphs

Review articles

Jens Bolte and Sebastian Endres  Trace formulae for quantum graphs [ MR 2459873 ]

J. M. Harrison  Quantum graphs with spin Hamiltonians [ MR 2459874 ]

J. P. Keating  Quantum graphs and quantum chaos [ MR 2459875 ]

Peter Kuchment  Quantum graphs: An introduction and a brief survey [ MR 2459876 ]

Research articles

G. Berkolaiko  Two constructions of quantum graphs and two types of spectral statistics [ MR 2459877 ]

B. M. Brown, M. S. P. Eastham and I. G. Wood  An example on the discrete spectrum of a star graph [ MR 2459878 ]

B. Malcolm Brown, Matthias Langer and Karl Michael Schmidt  The HELP inequality on trees [ MR 2459879 ]

Robert Carlson  Boundary value problems for infinite metric graphs [ MR 2459880 ]

Tomas Ekholm, Rupert L. Frank and Hynek Kovařík  Remarks about Hardy inequalities on metric trees [ MR 2459881 ]

Holger Flechsig and Sven Gnutzmann  On the spectral gap in Andreev graphs [ MR 2459882 ]

G. Freiling, M. Ignatiev and V. Yurko  An inverse spectral problem for SturmLiouville operators with singular potentials on startype graphs [ MR 2459883 ]

Michael J. Gruber, Mario Helm and Ivan Veselić  Optimal Wegner estimates for random Schrödinger operators on metric graphs [ MR 2459884 ]

Vadim Kostrykin, Jürgen Potthoff and Robert Schrader  Contraction semigroups on metric graphs [ MR 2459885 ]

Konstantin Pankrashkin  Localization in a quasiperiodic model on quantum graphs [ MR 2459886 ]

Olaf Post  Equilateral quantum graphs and boundary triples [ MR 2459887 ]

B. Winn  A conditionally convergent trace formula for quantum graphs [ MR 2459888 ]

Chapter 4. Applications

Review articles

Sergei Avdonin  Control problems on quantum graphs [ MR 2459889 ]

Pavel Exner  Leaky quantum graphs: A review [ MR 2459890 ]

Daniel Grieser  Thin tubes in mathematical physics, global analysis and spectral geometry [ MR 2459891 ]

Oleh Hul, Michał Ławniczak, Szymon Bauch and Leszek Sirko  Simulation of quantum graphs by microwave networks [ MR 2459892 ]

David Krejčiřík  Twisting versus bending in quantum waveguides [ MR 2459893 ]

Research articles

B. Bellazzini, M. Burrello, M. Mintchev and P. Sorba  Quantum field theory on star graphs [ MR 2459894 ]

H. D. Cornean, P. Duclos and B. Ricaud  On the skeleton method and an application to a quantum scissor [ MR 2459895 ]

S. A. Fulling and J. H. Wilson  Vacuum energy and closed orbits in quantum graphs [ MR 2459896 ]

Philipp Schapotschnikow and Sven Gnutzmann  Spectra of graphs and semiconducting polymers [ MR 2459897 ]


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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 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.
Graduate students and research mathematicians interested in various areas of analysis, dynamical systems, groups and their actions on graphs, mathematical physics.

Chapter 1. Analysis on combinatorial graphs

Review articles

Ram Band, Idan Oren and Uzy Smilansky  Nodal domains on graphs—how to count them and why? [ MR 2459862 ]

Matthew D. Horton, H. M. Stark and Audrey A. Terras  Zeta functions of weighted graphs and covering graphs [ MR 2459863 ]

Toshikazu Sunada  Discrete geometric analysis [ MR 2459864 ]

Research articles

Michael J. Gruber, Daniel H. Lenz and Ivan Veselić  Uniform existence of the integrated density of states for combinatorial and metric graphs over $\mathbb {Z}^d$ [ MR 2459865 ]

Daniele Guido, Tommaso Isola and Michel L. Lapidus  Bartholdi zeta functions for periodic simple graphs [ MR 2459866 ]

M. Kelbert and Y. Suhov  Asymptotic properties of Markov processes on Cayley trees [ MR 2459867 ]

Chapter 2. Analysis on fractals

Review articles

Volodymyr Nekrashevych and Alexander Teplyaev  Groups and analysis on fractals [ MR 2459868 ]

Research articles

Rostislav Grigorchuk and Zoran Šunić  Schreier spectrum of the Hanoi Towers group on three pegs [ MR 2459869 ]

Alexander Grigor′yan and Takashi Kumagai  On the dichotomy in the heat kernel two sided estimates [ MR 2459870 ]

Michel L. Lapidus and Erin P. J. Pearse  Tube formulas for selfsimilar fractals [ MR 2459871 ]

Roberto Peirone  Existence of eigenforms on nicely separated fractals [ MR 2459872 ]

Chapter 3. Analysis on quantum graphs

Review articles

Jens Bolte and Sebastian Endres  Trace formulae for quantum graphs [ MR 2459873 ]

J. M. Harrison  Quantum graphs with spin Hamiltonians [ MR 2459874 ]

J. P. Keating  Quantum graphs and quantum chaos [ MR 2459875 ]

Peter Kuchment  Quantum graphs: An introduction and a brief survey [ MR 2459876 ]

Research articles

G. Berkolaiko  Two constructions of quantum graphs and two types of spectral statistics [ MR 2459877 ]

B. M. Brown, M. S. P. Eastham and I. G. Wood  An example on the discrete spectrum of a star graph [ MR 2459878 ]

B. Malcolm Brown, Matthias Langer and Karl Michael Schmidt  The HELP inequality on trees [ MR 2459879 ]

Robert Carlson  Boundary value problems for infinite metric graphs [ MR 2459880 ]

Tomas Ekholm, Rupert L. Frank and Hynek Kovařík  Remarks about Hardy inequalities on metric trees [ MR 2459881 ]

Holger Flechsig and Sven Gnutzmann  On the spectral gap in Andreev graphs [ MR 2459882 ]

G. Freiling, M. Ignatiev and V. Yurko  An inverse spectral problem for SturmLiouville operators with singular potentials on startype graphs [ MR 2459883 ]

Michael J. Gruber, Mario Helm and Ivan Veselić  Optimal Wegner estimates for random Schrödinger operators on metric graphs [ MR 2459884 ]

Vadim Kostrykin, Jürgen Potthoff and Robert Schrader  Contraction semigroups on metric graphs [ MR 2459885 ]

Konstantin Pankrashkin  Localization in a quasiperiodic model on quantum graphs [ MR 2459886 ]

Olaf Post  Equilateral quantum graphs and boundary triples [ MR 2459887 ]

B. Winn  A conditionally convergent trace formula for quantum graphs [ MR 2459888 ]

Chapter 4. Applications

Review articles

Sergei Avdonin  Control problems on quantum graphs [ MR 2459889 ]

Pavel Exner  Leaky quantum graphs: A review [ MR 2459890 ]

Daniel Grieser  Thin tubes in mathematical physics, global analysis and spectral geometry [ MR 2459891 ]

Oleh Hul, Michał Ławniczak, Szymon Bauch and Leszek Sirko  Simulation of quantum graphs by microwave networks [ MR 2459892 ]

David Krejčiřík  Twisting versus bending in quantum waveguides [ MR 2459893 ]

Research articles

B. Bellazzini, M. Burrello, M. Mintchev and P. Sorba  Quantum field theory on star graphs [ MR 2459894 ]

H. D. Cornean, P. Duclos and B. Ricaud  On the skeleton method and an application to a quantum scissor [ MR 2459895 ]

S. A. Fulling and J. H. Wilson  Vacuum energy and closed orbits in quantum graphs [ MR 2459896 ]

Philipp Schapotschnikow and Sven Gnutzmann  Spectra of graphs and semiconducting polymers [ MR 2459897 ]