H. Ammari and F. Santosa, Guided waves in a photonic bandgap structure with a line defect, SIAM J. Appl. Math, vol.64, issue.6, pp.2018-2033, 2004.

Y. Avishai and J. M. Luck, Quantum percolation and ballistic conductance on a lattice of wires, Physical Review B, vol.45, issue.3, p.1074, 1992.
DOI : 10.1103/PhysRevB.45.1074

F. L. Bakharev, S. A. Nazarov, and K. M. Ruotsalainen, A gap in the spectrum of the Neumann???Laplacian on a periodic waveguide, Applicable Analysis, vol.32, issue.5, pp.1889-1915, 2013.
DOI : 10.1080/00036810903479715

A. Bendali, A. Huard, A. Tizaoui, S. Tordeux, and J. P. Vila, Asymptotic expansions of the eigenvalues of a 2-D boundary-value problem relative to two cavities linked by a hole of small size, C. R. Math. Acad. Sci. Paris, vol.347, pp.19-201147, 2009.
URL : https://hal.archives-ouvertes.fr/inria-00527437

G. Berkolaiko and P. Kuchment, Introduction to quantum graphs, volume 186 of Mathematical Surveys and Monographs

M. Sh, M. Z. Birman, and . Solomjak, Spectral theory of selfadjoint operators in Hilbert space Mathematics and its Applications (Soviet Series). D, 1987.

G. Borg, Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe: Bestimmung der Differentialgleichung durch die Eigenwerte, Acta Mathematica, vol.78, issue.0, pp.1-96, 1946.
DOI : 10.1007/BF02421600

R. Carlson, Adjoint and self-adjoint differential operators on graphs, Electronic Journal of Differential Equations, vol.6, pp.1-10, 1998.

M. S. Eastham, The spectral theory of periodic differential equations, 1973.

P. Exner, Lattice kronig-penney models. Physical review letters, p.3503, 1995.

P. Exner, Contact interactions on graph superlattices, Journal of Physics A: Mathematical and General, vol.29, issue.1, p.87, 1996.
DOI : 10.1088/0305-4470/29/1/011

A. Figotin and A. Klein, Localized classical waves created by defects, Journal of Statistical Physics, vol.18, issue.6, pp.165-177, 1997.
DOI : 10.1007/BF02180202

A. Figotin and A. Klein, Midgap defect modes in dielectric and acoustic media, SIAM J. Appl. Math, vol.58, issue.6, pp.1748-1773, 1998.

A. Figotin and P. Kuchment, Band-Gap Structure of Spectra of Periodic Dielectric and Acoustic Media. I. Scalar Model, SIAM Journal on Applied Mathematics, vol.56, issue.1, pp.68-88, 1996.
DOI : 10.1137/S0036139994263859

A. Figotin and P. Kuchment, Band-Gap Structure of Spectra of Periodic Dielectric and Acoustic Media. II. Two-Dimensional Photonic Crystals, SIAM Journal on Applied Mathematics, vol.56, issue.6, pp.1561-1620, 1996.
DOI : 10.1137/S0036139995285236

A. M. Il-in, Matching of asymptotic expansions of solutions of boundary value problems, volume 102 of Translations of Mathematical Monographs, 1992.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal -Molding the Flow of Light, 1995.

S. G. Johnson and J. D. Joannopoulos, Photonic Crystal -The road from theory to practice, 2002.

P. Joly and A. Semin, Construction and analysis of improved Kirchoff conditions for acoustic wave propagation in a junction of thin slots, Paris-Sud Working Group on Modelling and Scientific Computing, pp.44-67, 2007.
DOI : 10.1051/proc:082504

URL : https://hal.archives-ouvertes.fr/hal-00976404

P. Joly and S. Tordeux, Matching of Asymptotic Expansions for Wave Propagation in Media with Thin Slots I: The Asymptotic Expansion, Multiscale Modeling & Simulation, vol.5, issue.1, pp.304-336, 2006.
DOI : 10.1137/05064494X

URL : https://hal.archives-ouvertes.fr/inria-00527588

P. Joly and S. Tordeux, Matching of asymptotic expansions for waves propagation in media with thin slots II: The error estimates, ESAIM: Mathematical Modelling and Numerical Analysis, vol.42, issue.2, pp.193-221, 2008.
DOI : 10.1051/m2an:2008004

URL : https://hal.archives-ouvertes.fr/inria-00527445

P. Kuchment, Floquet theory for partial differential equations, volume 60 of Operator Theory: Advances and Applications, 1993.

P. Kuchment and B. Ong, On guided waves in photonic crystal waveguides, Waves in periodic and random media, pp.105-115, 2002.
DOI : 10.1090/conm/339/06102

P. Kuchment, Quantum graphs. I. Some basic structures. Waves Random Media, pp.107-128, 2004.

P. Kuchment, Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs, Journal of Physics A: Mathematical and General, vol.38, issue.22, pp.4887-4900, 2005.
DOI : 10.1088/0305-4470/38/22/013

P. Kuchment, Quantum graphs, of Proc. Sympos. Pure Math, pp.291-312, 2008.
DOI : 10.1088/0959-7174/14/1/007

P. Kuchment and B. Ong, On guided electromagnetic waves in photonic crystal waveguides In Operator theory and its applications, volume 231 of Amer, Math. Soc. Transl. Ser, vol.2, pp.99-108, 2010.

P. Kuchment and Y. Pinchover, Integral Representations and Liouville Theorems for Solutions of Periodic Elliptic Equations, Journal of Functional Analysis, vol.181, issue.2, pp.402-446, 2001.
DOI : 10.1006/jfan.2000.3727

P. Kuchment and H. Zeng, Convergence of Spectra of Mesoscopic Systems Collapsing onto a Graph, Journal of Mathematical Analysis and Applications, vol.258, issue.2, pp.671-700, 2001.
DOI : 10.1006/jmaa.2000.7415

V. Maz-'ya, S. Nazarov, and B. Plamenevskij, Asymptotic theory of elliptic boundary value problems in singularly perturbed domains of Operator Theory: Advances and Applications, 2000.

W. Mclean, Strongly elliptic systems and boundary integral equations, 2000.

S. A. Nazarov, An example of multiple gaps in the spectrum of a periodic waveguide, Sbornik: Mathematics, vol.201, issue.4, pp.99-124, 2010.
DOI : 10.1070/SM2010v201n04ABEH004082

S. A. Nazarov, Opening of a gap in the continuous spectrum of a periodically perturbed waveguide, Mathematical Notes, vol.87, issue.5-6, pp.764-786, 2010.
DOI : 10.1134/S0001434610050123

S. A. Nazarov, The asymptotic analysis of gaps in the spectrum of a waveguide perturbed with a periodic family of small voids, Journal of Mathematical Sciences, vol.22, issue.1, pp.247-301, 2012.
DOI : 10.1007/s10958-012-0985-y

L. Parnovski, Bethe???Sommerfeld Conjecture, Annales Henri Poincar??, vol.9, issue.3, pp.457-508, 2008.
DOI : 10.1007/s00023-008-0364-x

L. Parnovski and A. V. Sobolev, Bethe-Sommerfeld conjecture for periodic operators with strong perturbations, Inventiones mathematicae, vol.2, issue.2, pp.467-540, 2010.
DOI : 10.1007/s00222-010-0251-1

O. Post, Spectral Convergence of Quasi-One-Dimensional Spaces, Annales Henri Poincar??, vol.7, issue.5, pp.933-973, 2006.
DOI : 10.1007/s00023-006-0272-x

O. Post, Spectral analysis on graph-like spaces, Lecture Notes in Mathematics, vol.2039
DOI : 10.1007/978-3-642-23840-6

M. Reed and B. Simon, Methods of modern mathematical physics v. I-IV, pp.1972-1978

J. Rubinstein and M. Schatzman, Variational Problems??on Multiply Connected Thin Strips I:??Basic Estimates and Convergence??of the Laplacian Spectrum, Archive for Rational Mechanics and Analysis, vol.160, issue.4, pp.271-308, 2001.
DOI : 10.1007/s002050100164

Y. Saito, The limiting equation for Neumann Laplacians on shrinking domains, Electron . J. Differential Equations, vol.25, issue.31, p.pp, 2000.

A. Semin, Propagation d'ondes dans des jonctions de fentes minces, 2010.

M. Van-dyke, Perturbation Method in Fluid Mechanics, Journal of Applied Mechanics, vol.43, issue.1, 1964.
DOI : 10.1115/1.3423785

M. Vorobets, On the Bethe???Sommerfeld conjecture for certain periodic Maxwell operators, Journal of Mathematical Analysis and Applications, vol.377, issue.1, pp.370-383, 2011.
DOI : 10.1016/j.jmaa.2010.10.067