Analysis on Graphs and its Applications

by Exner, Pavel; Keating, Jonathan P.; Kuchment, Peter; Sunada, Toshikazu; Teplyaev, Alexander

• ISBN13: 9780821844717
• ISBN10: 0821844717
• Format: Hardcover
• Copyright: 2008-09-30
• Publisher: Amer Mathematical Society

Summary

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 wiresystems, 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.

Table of Contents

Analysis on combinatorial graphs: Review articles: Nodal domains on graphs–How to count them and why?

Zeta functions of weighted graphs and covering graphs

Discrete geometric analysis

Research articles: Uniform existence of the integrated density of states for combinatorial and metric graphs over $Z^d$

Bartholdi zeta functions for periodic simple graphs

Asymptotic properties of Markov processes on Cayley trees

Analysis on fractals: Review articles: Groups and analysis on fractals

Research articles: Schreier spectrum of the Hanoi Towers group on three pegs

On the dichotomy in the heat kernel two sided estimates

Tube formulas for self-similar fractals

Existence of Eigenforms on nicely separated fractals

Analysis on quantum graphs: Review articles: Trace formulae for quantum graphs

Quantum graphs with spin Hamiltonians

Quantum graphs and quantum chaos

Quantum graphs: An Introduction and a brief survey

Research articles: Two constructions of quantum graphs and two types of spectral statistics

An example on the discrete spectrum of a star graph

The HELP inequality on trees

Boundary value problems for infinite metric graphs

Remarks about Hardy inequalities on metric trees

On the spectral gap in Andreev graphs

An inverse spectral problem for Sturm-Liouville operators with singular potentials on star-type graphs

Optimal Wegner estimates for random Schrodinger operators on metric graphs

Contraction semigroups on metric graphs

Localization in a quasiperiodic model on quantum graphs

Equilateral quantum graphs and boundary triples

A conditionally convergent trace formula for quantum graphs

Applications: Review articles: Control problems on quantum graphs

Leaky quantum graphs: A review

Thin tubes in mathematical physics, global analysis and spectral geometry

Simulation of quantum graphs by microwave networks

Twisting versus bending in quantum waveguides

Research articles: Quantum field theory on star graphs

On the skeleton method and an application to a quantum scissor

Vacuum energy and closed orbits in quantum graphs

Spectra of graphs and semi-conducting polymers

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