Graph Theory in Physics  

In General > s.a. graph theory / quantum systems.
@ General references: Estrada a1302-ch [introduction]; Jouneghani et al IJTP(14)-a1309 [review of quantum graphical models].
@ Lagrangian systems: Novikov & Schvarts RMS(99)mp/00.
@ Discretized field theories: Kan & Shiraishi JMP(05)ht/04 [QED, divergences]; > s.a. lattice field theory.
@ Quantum mechanics on graphs: Ettinger & Hoyer qp/99 [graph isomorphisms]; Barra & Gaspard PRE(02)cm/01; Blümel et al qp/02 [regular, mathematical foundations]; Bolte & Harrison JPA(03) [form factor, spin]; Błasiak & Horzela a0710 ≠ Błasiak et al JPCS(10) [graph operator algebras]; Pavičić et al JMP(10)-a1004 [graph approach to quantum systems]; Harrison et al PRS(11) [particle statistics]; Mintchev JPA(11)-a1106 [non-equilibrium steady states on star graphs]; Ionicioiu & Spiller PRA(12)-a1110 [mapping graphs to quantum states]; > s.a. cellular automaton; spin models [graph states].
@ Graph evolution models: Dadic & Pisk IJTP(79); Markopoulou & Prémont-Schwarz CQG(08)-a0805 [conserved topological defects]; Rath & Toth EJP(09)-a0808 [random graphs and self-organized critical state]; Grindrod & Higham PRS(10); Arrighi & Dowek a1202 [causal dynamics]; Chen & Plotkin PRD(13)-a1210 [and emergent manifolds]; Arrighi & Martiel a1607 [Quantum Causal Graph Dynamics].
@ Related topics: Barra & Gaspard PRE(01) [classical dynamics]; Fiorenza ACS(06)m.CT/02 [sums over graphs]; Giorda & Zanardi PRA(03)qp, EPL(04)qp/03 [bosonic, entanglement and tunneling]; Procacci & Scoppola CPAA-mp/05 [random cluster model]; Reidys DM(08) [sequential dynamical systems]; Bombelli et al CQG(09)-a0905 [manifolds from graphs in lqg]; Cornelissen & Marcolli JGP(13) [graph reconstruction and quantum statistical mechanics]; Cabello et al PRL(14)-a1401 [graph invariants and quantum correlations]; > s.a. entangled systems [graph states]; Virial Expansion.
> Specific models: see Anderson Localization; discrete geometry models [gravity, quantum graphity]; Polymers; supersymmetric theories; toda lattice.

Statistical Models > s.a. networks [entropy]; game theory; phase transitions; Power-Law Distribution; stochastic processes.
@ Random walks: Watrous cs.CC/98-in; Burioni & Cassi JPA(05) [rev]; Mendonça PRE(11)-a1106 [cover time]; Arendt & Jost a1203/EPJB [multiple-agent consensus problem]; > s.a. diffusion; green functions.
@ Quantum walks: Farhi & Gutmann PRA(98); Aharonov et al qp/00-proc; Kendon IJQI(06)qp/03 [discrete time]; Montanaro QIC(07)qp/05; Osborne PRL(08)qp/06 [approximate locality]; Kargin JPA(10) [bounds on the mixing time]; Higuchi et al JMI-a1207 [discrete-time evolution operators].
@ Transport: Muelken & Blumen PRE(06)qp [quantum vs classical percolability].
@ Random graphs, evolution: Barbosa et al PhyA(04) [directed]; Lee et al NPB(04) [as Potts model]; Lushnikov JPA(05); Turova JSP(06) [phase transitions]; > s.a. ising model; networks.
@ Quantum-gravity motivated: Finkel ht/06-conf [local moves and lqg]; Conrady JSP(11)-a1009 [emergence of 2D space]; > s.a. discrete geometry.
@ Fields on graphs: Häggström AAP(00) [percolation, phase transitions]; Gobron a1312 [Pfaffian representations of Ising partition function, non-planar graphs]; > s.a. entanglement entropy; wave equation.
@ Thermodynamics on graphs: Burioni et al JPA(00) [spectral partitions into subgraphs]; Majka & Wislicki PhyA(04) [communication networks].
@ Quantum field theory on graphs: Cimasoni & Reshetikhin CMP(08)-a0704 [from dimer model].

Operators
@ General references: Requardt mp/00/JPA [spectral analysis and Connes distance]; Kostrykin & Schrader JMP(01)mp/00 [scattering matrices].
@ Laplacian: Forman Top(93) [determinant]; Akkermans et al AP(00) [spectral determinant]; Khorunzhy & Vengerovsky mp/00 [random graph]; Requardt JPA(02)mp/01 [Dirac operator and Connes metric]; Dean JPA(02) [density of states]; Kenyon mp/02; Hashimoto et al JMP(03) [large graph, spectral distribution]; Dorogovtsev et al PhyA(04) [random, spectrum]; Braunstein et al AC(06)qp/04 [as a density matrix]; Khorunzhiy et al AAP(06)mp/05 [random graph, tails of spectra]; Müller & Stollmann JFA(07)mp/05 [on supercritical bond-percolation graphs]; Kostrykin & Schrader mp/06 [inverse scattering]; Hu DM(07) [eigenvalues, and adjacency matrix]; Elon JPA(08)-a0804 [statistical approach]; Schrader JPA(09) [Klein-Gordon and wave equation]; Keller & Lenz a1101; Haeseler et al a1103 [infinite graphs]; Bauer et al a1211 [spectrum]; Majid JGP(13) [non-commutative geometry and canonical edge Laplacian]; Anantharaman a1512 [on large graphs, quantum ergodicity]; > s.a. types of graphs [random].
@ Laplacian, bounds on eigenvalues: Das & Guo DM(13) [for the second power of a graph]; Charles et al DM(13) [and non-positive eigenvalues of the adjacency matrix].
@ Schrödinger operator: Novikov RMS(97)mp/00; Kostrykin & Schrader JPA(00)mp, RVMP(00)mp [1D]; Gutkin & Smilansky JPA(01) [the spectrum determines the graph uniquely].
@ Other operators: Bolte & Harrison JPA(03) [Dirac operator, spectral statistics]; Exner a1205-fs [momentum operators].

Quantum Graphs
* Idea: A quantum graph is a graph considered as a (singular) one-dimensional variety and equipped with a second-order differential Hamiltonian H (a "Laplacian") with suitable conditions at the vertices; They are commonly used as models of complex quantum systems, for example molecules, networks of wires, and states of condensed matter.
@ General references: Dabaghian & Blümel PRE(03)qp, JETPL(03)qp, qp/03 [analytically solvable]; Schmidt et al JPA(03) [Green functions]; Kurasov & Nowaczyk JPA(05) [inverse spectral problem]; Fulling m.SP/05-conf [local spectral density and vacuum energy]; Exner et al RVMP(07) [random potential, localization]; Fulling et al JPA(07)-a0708 [index theorems]; Kuchment in(08)-a0802 [rev]; Weaver a1506 [as quantum relations]; Andrade et al PRP(16)-a1601 [Green's function approach]; Berkolaiko a1603 [elementary intro].
@ And other physics: Dabaghian et al JETPL(01)qp [and chaos]; Tanner qp/05 [and quantum random walks].
@ Spectral properties: Barra & Gaspard JSP(00)qp [level spacing distribution]; Kuchment JPA(05)mp/04; Gavish & Smilansky JPA(07)-a0807 [spectral theory and length spectrum]; Berkolaiko & Liu a1601 [simplicity of eigenvalues and non-vanishing of eigenfunctions].


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