{X,Y}-Cyclacene Graphs with Next Nearest Neighbor Interactions

References

• Choi, H. S. and K. S., Kim. “Structures, Magnetic Properties, and Aromaticity of Cyclacenes.” Angew. Chem. Int. Edition 38 (1999): 2256–58.
   PubMed Web of Science ®Google Scholar
• Nakamura, E., K. Tahara, Y. Matsuo, and M. Sawamura. “Synthesis, Structure, and Aromaticity of a Hoop-Shaped Cyclic Benzenoid [10] Cyclophenacene.” J. Am. Chem. Soc. 15 (2003): 2834–5.
   Google Scholar
• Houk, K. N., P. S. Lee, and M. Nendel. “Polyacene and Cyclacene Geometries and Electronic Structures: Bond Equalization, Vanishing Band Gaps, and Triplet Ground States Contrast with Polyacetylene.” J. Org. Chem. 66 (2001): 5517–21.
   PubMed Web of Science ®Google Scholar
• Tahara, K., and Y. Tobe. “Molecular Loops and Belts.” Chem. Rev. 106 (2006): 5279–90.
   Google Scholar
• Saito, Y. and M. Maida. “Square, Pentagon, and Heptagon Rings at BN Nanotube Tips.” J. Phys. Chem. A 103 (1999): 1291–3.
   Web of Science ®Google Scholar
• Zope, R. R., and B. I. Dunlap. “Are Hemispherical Caps of Boron–Nitride Nanotubes Possible?” Chem. Phys. Lett. 386 (2004): 403–7.
   Web of Science ®Google Scholar
• Mandal, B. “Graph Theoretical Procedure for Obtaining Analytical Expressions of Eigenspectra of Linear Chains and Cycles with Alternant Vertex Weights and Same Edge Weight: Application to Some Complicated Graphs.” Int. J. Quantum Chem. 103 (2005): 140–8.
   Web of Science ®Google Scholar
• Bej, N., D. C. Mukherjee, and A. K. Mukherjee. “Eigenspectra and Characteristic Polynomials of Cylindrical Polyacene Strips.” Mol. Phys. 101 (2003): 865–70.
   Web of Science ®Google Scholar
• Klein, D. J. “Hückel-Model Solution for Polyphenanthrene Strips.” Rep. Molec. Theor. 1 (1990): 91–4.
   Google Scholar
• Dias, J. R. “Strongly Subspectral Conjugated Molecular Systems. From Small Molecules to Infinitely Large π-Electronic Networks.” J. Phys. Chem. A 101 (1997): 7167–75.
   Web of Science ®Google Scholar
• Hosoya, H., H. Kumazaki, K. Chida, M. Ohuchi, and Y., Gao “How do the Polycyclic Aromatic Hydrocarbons Approach Infinity?.” Pure Appl. Chem. 62 (1990): 445–50.
   Web of Science ®Google Scholar
• Ghosh, P., D. J. Klein, and B. Mandal. “Analytical Eigenspectra of Alternant Edge-Weighted Graphs of Linear Chains and Cycles: Some Applications.” Mol. Phys. 112 (2014): 2093–2106.
   Web of Science ®Google Scholar
• Karmakar, S., S. Mondal, and B. Mandal. “Eigensolutions of Cycloployacene Graphs.” Mol. Phys. 113 (2015): 719–26.
   Web of Science ®Google Scholar
• Türker, L., and S. Gümüs. “Cyclacenes.” J. Mol. Struct. (THEOCHEM) 685 (2004): 1–33.
   Web of Science ®Google Scholar
• Trinajstić, N. Chemical Graph Theory. (Boca Raton: CRC Press, 1992).
   Google Scholar
• Dias, J. R. Molecular Orbital Calculations Using Chemical Graph Theory. (Berlin: Springer-Verlag, 1993).
   Google Scholar
• Gutman, I., and O. E. Polansky. Mathematical Concepts in Organic Chemistry. (Berlin: Springer-Verlag, 1986).
   Google Scholar
• McClelland, B. J. “Graphical Method for Factorizing Secular Determinants of Hückel Molecular Orbital Theory.” J. Chem. Soc. Faraday Trans. 2 (70) (1974): 1453–6.
   Google Scholar
• McClelland, B. J. “Eigenvalues of the Topological Matrix. Splitting of Graphs with Symmetrical Components and Alternant Graphs.” J. Chem. Soc. Faraday Trans. 2 (78) (1982): 911–6.
   Google Scholar
• McClelland, B. J. “On the Graphical Factorization of Hückel Characteristic Equations.” Mol. Phys. 45 (1982): 189–190.
   Web of Science ®Google Scholar
• Dias, J. R. “A Facile Huckel Molecular Orbital Solution of Buckminsterfullerene Using Chemical Graph Theory.” J. Chem. Educ. 66 (1989): 1012-.
   Web of Science ®Google Scholar
• D'Amato, S. S. “Eigenvalues of Graphs with Twofold Symmetry.” Mol. Phys. 37 (1979): 1363–9.
   Web of Science ®Google Scholar
• D'Amato, S. S. “Eigenvalues of Graphs with Threefold Symmetry.” Theor. Chim. Acta 53 (1979): 319–26.
   Google Scholar
• Davidson, R. A. “Spectral Analysis of Graphs by Cyclic Automorphism Subgroups.” Theor. Chim. Acta 58 (1981): 193–231.
   Google Scholar
• Shen, M. “Splitting of n-Fold Rotationally Symmetric Graphs.” Int. J. Quantum Chem. 38 (1990): 551–7.
   Web of Science ®Google Scholar
• Hosoya, H., and Y. Tsukano. “Efficient Way for Factorizing the Characteristic Polynomial of Highly Symmetrical Graphs Such as the Buckminsterfullerene.” Fullerene Sci. Technol. 2 (1994): 381–93.
   Web of Science ®Google Scholar
• Hosoya, H., T. Tsukano, K. Nakada, S. Iwata, and U. Nagashima. “Supersymmetry of Hexabenzocoronene Torus.” Croat. Chem. Acta 77 (2004): 89–95.
   Web of Science ®Google Scholar
• Dias, J. R. “Techniques in Facile Calculation of Molecular Orbital Parameters and Related Conceptualizations—Molecular Orbital Functional Groups.” J. Mol. Struct. (THEOCHEM) 417 (1997): 49–67.
   Web of Science ®Google Scholar
• Dias, J. R. “Properties and Relationships of Right-Hand Mirror-Plane Fragments and Their Eigenvectors: the Concept of Complementarity of Molecular Graphs Molecular Orbital Functional Groups- Part 2.” Mol. Phys. 88 (1996): 407–17.
   Web of Science ®Google Scholar
• Coulson, C. A., and G. S. Rushbrooke. “Note on the Method of Molecular Orbitals.” Proc. Cambridge Philos. Soc. 36 (1940): 193–200.
   Google Scholar
• Mukherjee, A. K., and D. K. Das “Graph Factorization: A New Mode of Application of a Vertex-Alternation Scheme.” Int. J. Quantum. Chem. 46 (1993): 519–33.
   Web of Science ®Google Scholar
• Kassman, A. J. “Generation of the Eigenvectors of the Topological Matrix from Graph Theory.” Theor. Chim. Acta (Berl.) 67 (1985): 255–62.
   Google Scholar
• Mukherjee, A. K., and K. K. Dutta. “Two New Graph-Theoretical Methods for Generation of Eigenvectors of Chemical Graphs.” Proc. Indian Acad. Sci. 101 (1989): 499–517.
   Google Scholar
• Mandal, B. “Use of Symmetry Plane and Subsequent Subtraction for Obtaining Eigenspectra of Some Complicated Graphs in Analytical Forms.” J. Mol. Struct. (THEOCHEM) 757 (2005): 99–111.
   Web of Science ®Google Scholar
• Mandal, B. “Eigenspectral Analysis of Pendant Vertex- and Pendant Edge-Weighted Graphs of Linear Chains, Cycles, and Stars.” Bull. Chem. Soc. Jpn. 81 (2008): 956–65.
   Web of Science ®Google Scholar
• Mandal, B., and D. J. Klein. “Characteristic Polynomial Followed by Trigonometric Identity for Obtaining Analytical Eigenspectra of Some Weighted Graphs of Linear Chains and Cycles.” Bull. Chem. Soc. Jpn. 87 (2014): 491–97.
   Web of Science ®Google Scholar
• Hall, G. G. “Some solutions in the Graph Theory of Alternant Benzenoids.” Int. J. Quantum Chem. 45 (1993): 167–76.
   Web of Science ®Google Scholar
• Hall, G. G., and S. Arimoto. “Eigenvalue Distributions and Asymptotic Lines of the Energy in Alternant Hydrocarbons.” Int. J. Quantum Chem. 45 (1993): 303–28.
   Web of Science ®Google Scholar
• Karmakar, S., and B. Mandal. “Graph Theoretical Solutions for the Coupled Kinetic Rate Equations.” J. Phys. Chem. A 118 (2014): 1155–61.
   PubMed Web of Science ®Google Scholar
• Karmakar, S., and B. Mandal. “Graph Theoretical Analysis on the Kinetic Rate Equations of Linear Chain and Cyclic Reaction Networks.” J. Phys. Chem. A 118 (2014): 7672–82.
   PubMed Web of Science ®Google Scholar
• Ghosh, P., S. Karmakar, and B. Mandal. “Eigensolutions of Dodecahedron Graphs.” Chem. Phys. Lett. 594 (2014): 41–6.
   Web of Science ®Google Scholar
• Klein, D. J., and X. Liu. “Theorems for Carbon Cages.” J. Math. Chem. 11 (1992): 199–205.
   Web of Science ®Google Scholar
• Klein, D. J., and B. Mandal “Local Symmetries for Molecular Graphs.” MATCH Commun. Math. Comput. Chem. 74 (2015): 247–58.
   Web of Science ®Google Scholar
• Van-Catledge, F. A. “A Pariser-Parr-Pople-Based Set of Hueckel Molecular Orbital Parameters.” J. Org. Chem. 45 (1980): 4801–2.
   Web of Science ®Google Scholar
• Gutman, I., and O. E. Polansky. “Cyclic Conjugation and the Hückel Molecular Orbital Model.” Theor. Chim. Acta 60 (1981): 203–26.
   Google Scholar
• Balasubramanian, K. “Characteristic Polynomials of Organic Polymers and Periodic Structures.” J. Comput. Chem. 6 (1985): 656–61.
   Web of Science ®Google Scholar
• King, R. B. “Chemical Applications of Group Theory and Topology.” Theor. Chim. Acta 56 (1980): 269–96.
   Google Scholar
• Klein, D. J. “Graph Theoretically Formulated Electronic-Structure Theory.” Internet Electronic. J. Mol. Des. 2 (2003): 814–34.
   Google Scholar
• Hubbard, J. “Electron Correlations in Narrow Energy Bands.” Proc. Royal Soc. (London) A 276 (1963): 238–57.
   Web of Science ®Google Scholar
• Lieb, E. H. “Two Theorems on the Hubbard Model.” Phys. Rev. Lett. 62 (1989): 1201–4.
   PubMed Web of Science ®Google Scholar
• Zaleski, T. A., and T. K. Kopeć. “Effect of Next-Nearest-Neighbour Hopping on Bose–Einstein Condensation in Optical Lattices.” J. Phys. B: At. Mol. Opt. Phys. 43 (2010): 85303.
   Web of Science ®Google Scholar
• Lieb, E., T. Schultz, and D. Mattis. “Two Soluble Models of an Antiferromagnetic Chain.” Anal. Phys. 16 (1961): 407–66.
   Web of Science ®Google Scholar
• Katsumata, K. “High-Frequency Electron Spin Resonance in Magnetic Systems.” J. Phys. Cond. Matter 12 (2000): R589–R614.
   Web of Science ®Google Scholar
• Majumdar, C. K., and D. K. Ghosh. “On Next‐Nearest‐Neighbor Interaction in Linear Chain. I & II.” J. Math. Phys. 10 (1969): 1388–1402.
   Web of Science ®Google Scholar