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Hardcover ISBN:  9780821844717 
Product Code:  PSPUM/77 
List Price:  $149.00 
MAA Member Price:  $134.10 
AMS Member Price:  $119.20 
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 
eBook ISBN:  9780821893845 
Product Code:  PSPUM/77.B 
List Price:  $288.00 $218.50 
MAA Member Price:  $259.20 $196.65 
AMS Member Price:  $230.40 $174.80 

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 ]


Additional Material

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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 ]