{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T05:15:44Z","timestamp":1765516544879,"version":"3.48.0"},"reference-count":141,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,12,10]],"date-time":"2025-12-10T00:00:00Z","timestamp":1765324800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let G=(V,E) be a simple graph with the vertex set V and the edge set E|V|=n,|E|=m. An example of a vertex-decorated graph DG is a vertex-quadrangulated graph QG. The vertex quadrangulation QG of 4-regular graph G visually looks like a graph whose vertices are depicted as empty squares, and the connecting edges are attached to the corners of the squares. If we contract each quadrangle of QG to a point that takes over the incidence of the four edges that were previously joined to this quadrangle, then we can again get the original graph G. Any connected graph H that provides (some of) its vertices for external connections can play the role of a decorating graph, and any graph G with vertices of valency no greater than the number of contact vertices in H can be decorated with it. Herein, we consider the case when G is a regular graph. Since the decoration also depends on the way the edges are attached to the decorating graph, we clearly stipulate it. We show that all similarly decorated regular graphs DG that meet our conditions have at least |V(H)| predicted common eigenvalues. A number of related results are proven. As possible applications of these results in chemistry, cases of simplified findings of eigenvalues of a molecular graph even in the absence of the usual symmetry of the molecule may be of interest. This, in particular, can somewhat expand the possibilities of applying the simple H\u00fcckel method for large molecules.<\/jats:p>","DOI":"10.3390\/axioms14120907","type":"journal-article","created":{"date-parts":[[2025,12,11]],"date-time":"2025-12-11T08:39:18Z","timestamp":1765442358000},"page":"907","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Common Eigenvalues of Vertex-Decorated Regular Graphs"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-8076-4710","authenticated-orcid":false,"given":"Vladimir R.","family":"Rosenfeld","sequence":"first","affiliation":[{"name":"Department of Mathematics, Ariel University, Ariel 4070000, Israel"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,10]]},"reference":[{"key":"ref_1","unstructured":"Balaban, A.T. (1976). Chemical Applications of Graph Theory, Academic Press."},{"key":"ref_2","unstructured":"Au-chin, T., Yuan-sun, K., Guo-sen, Y., and Shu-san, T. (1986). Graph Theoretical Molecular Orbitals, Science Press."},{"key":"ref_3","unstructured":"Papulov, Y.G., Rosenfeld, V.R., and Kemenova, T.G. (1990). Molecular Graphs, Tver University. (In Russian)."},{"key":"ref_4","unstructured":"Trinajsti\u0107, N. (1992). Chemical Graph Theory, CRC Press Taylor & Francis Group. [2nd ed.]."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Dehmer, M., and Emmert-Streib, F. (2015). Graph Theory: Mathematical Foundations and Applications, CRC Press Taylor & Francis Group.","DOI":"10.1002\/9783527693245"},{"key":"ref_6","unstructured":"Randi\u0107, M., Novi\u010d, M., and Plav\u0161i\u0107, D. (2016). Solved and Unsolved Problems of Structural Chemistry, CRC Press Taylor & Francis Group."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Temkin, O.N., Zeigarnik, A.V., and Bonchev, D. (2019). Chemical Reaction Networks, CRC Press Taylor & Francis Group.","DOI":"10.1201\/9781003067887"},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Wagner, S., and Wang, G. (2019). Introduction to Chemical Graph Theory, CRC Press Taylor & Francis Group.","DOI":"10.1201\/9780429450532"},{"key":"ref_9","unstructured":"Cvetkovi\u0107, D.M., Doob, M., and Sachs, H. (1980). Spectra of Graph\u2014Theory and Application, Academic Press."},{"key":"ref_10","unstructured":"Cvetkovi\u0107, D.M., Doob, M., Gutman, I., and Torga\u0161ev, A. (1988). Recent Results in the Theory of Graph Spectra, North Holland Publishing Company."},{"key":"ref_11","unstructured":"(2022, October 27). Spectral Graph Theory. Available online: https:\/\/en.wikiversity.org\/wiki\/Spectral_Graph_Theory."},{"key":"ref_12","unstructured":"Cvetkovi\u0107, D., and Gutman, I. (2011). Selected Topics on Applications of Graph Spectra, Matemati\u010dki Institut SANU."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"1545","DOI":"10.1016\/j.laa.2010.11.035","article-title":"Graph spectra in computer science","volume":"434","year":"2011","journal-title":"Linear Algebra Appl."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"1","DOI":"10.2298\/AADM111223025A","article-title":"Graph spectral techniques in computer sciences","volume":"6","year":"2012","journal-title":"Appl. Anal. Discrete Math."},{"key":"ref_15","first-page":"1","article-title":"Isospectral and subspectral molecules","volume":"34","author":"Gimarc","year":"1981","journal-title":"Croat. Chem. Acta"},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1007\/BF01170007","article-title":"On subspectral acyclic molecular graphs","volume":"4","author":"Jiang","year":"1990","journal-title":"J. Math. Chem."},{"key":"ref_17","first-page":"143","article-title":"On subspectral graphs","volume":"96","author":"Guo","year":"1993","journal-title":"Cong. Num."},{"key":"ref_18","doi-asserted-by":"crossref","unstructured":"Dias, J.R. (1993). Molecular Orbital Calculations Using Chemical Graph Theory, Springer.","DOI":"10.1007\/978-3-642-77894-0"},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"7167","DOI":"10.1021\/jp971552f","article-title":"Strongly subspectral conjugated molecular systems. From small molecules to infinitely large \u03c0-electronic networks","volume":"101","author":"Dias","year":"1997","journal-title":"J. Phys. Chem. A"},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"721","DOI":"10.1002\/(SICI)1097-461X(1999)74:6<721::AID-QUA12>3.0.CO;2-9","article-title":"Analysis of \u03c0-electronic structures of small alternant hydrocarbons to infinitely large polymeric strips: The Aufbau Principle and end-group effects","volume":"74","author":"Dias","year":"1999","journal-title":"Int. J. Quantum Chem."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"2922","DOI":"10.1039\/a902553g","article-title":"Strongly subspectral pairs in C50+10n and C60+12n fullerenes via a common generic graph","volume":"1","author":"Datta","year":"1999","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_22","first-page":"621","article-title":"p\u03c0-Molecular orbitals of conjugated linear polyene molecules as molecular orbital functional groups in the design of near-infrared dyes","volume":"75","author":"Dias","year":"2002","journal-title":"Croat. Chem. Acta"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1080\/00268976.2017.1395918","article-title":"Comprehensive study of the correlations that exhibit among the members of the [n]cyclacene series and the M\u00f6bius[n]cyclacene series","volume":"116","author":"Dias","year":"2018","journal-title":"Mol. Phys."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"339","DOI":"10.46793\/match.92-2.339K","article-title":"Eigen-persistence in graphs","volume":"92","author":"Klein","year":"2024","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1112\/blms\/3.3.321","article-title":"Cospectral graphs and digraphs","volume":"3","author":"Harary","year":"1971","journal-title":"Bull. Lond. Math. Soc."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/0040-4020(75)85002-2","article-title":"Isospectral graphs and molecules","volume":"31","author":"Herndon","year":"1975","journal-title":"Tetrahedron"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1017\/S0004972700007760","article-title":"A new graph product and its spectrum","volume":"18","author":"Godsil","year":"1978","journal-title":"Bull. Austral. Math. Soc."},{"key":"ref_28","first-page":"291","article-title":"Spectral properties of some graphs derived from bipartite graphs","volume":"8","author":"Gutman","year":"1980","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_29","first-page":"35","article-title":"Block and articulation node polynomials of the generalized rooted product of graphs","volume":"11","author":"Farrell","year":"2000","journal-title":"J. Math. Sci."},{"key":"ref_30","first-page":"142","article-title":"The block-polynomials and block-spectra of dendrimers","volume":"1","author":"Rosenfeld","year":"2002","journal-title":"Internet Electron. J. Mol. Des."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"500","DOI":"10.1016\/j.dam.2006.06.015","article-title":"The circuit polynomial of the restricted rooted product G(\u0393) of graphs with a bipartite core G","volume":"156","author":"Rosenfeld","year":"2008","journal-title":"Discrete Appl. Math."},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1016\/j.dam.2009.10.009","article-title":"The independence polynomial of rooted products of graphs","volume":"158","author":"Rosenfeld","year":"2010","journal-title":"Discrete Appl. Math."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"1526","DOI":"10.1007\/s10910-021-01250-6","article-title":"Close-to-zero eigenvalues of the rooted product of graphs","volume":"59","author":"Rosenfeld","year":"2021","journal-title":"J. Math. Chem."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"502","DOI":"10.1007\/s10910-021-01317-4","article-title":"The inertia and energy gap of a vertex-decorated graph with identically weighted \u2018internal\u2019 edges and beyond","volume":"60","author":"Rosenfeld","year":"2022","journal-title":"J. Math. Chem."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1023\/A:1011032212489","article-title":"The creation of spectral gaps by graph decoration","volume":"53","author":"Schenker","year":"2000","journal-title":"Lett. Math. Phys."},{"key":"ref_36","doi-asserted-by":"crossref","unstructured":"Harary, F. (1969). Graph Theory, Addison-Wesley Publishing Company.","DOI":"10.21236\/AD0705364"},{"key":"ref_37","unstructured":"Knill, O. (2014). A notion of graph homeomorphism. arXiv."},{"key":"ref_38","unstructured":"Mnuhin, V.B. (1980). Spectra of graphs under certain unary operations. Some Topological and Combinatorial Properties of Graphs, Institute of Mathematics. [1st ed.]. (In Russian)."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1019","DOI":"10.1039\/a708580j","article-title":"Eigenvalue relations for decorated trivalent polyhedra. Connections between the fullerenes and their fulleren-yne and spheriphane relatives","volume":"94","author":"Fowler","year":"1998","journal-title":"J. Chem. Soc. Faraday Trans."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"234114","DOI":"10.1063\/1.4883489","article-title":"Dirac cones in the spectrum of bond-decorated graphenes","volume":"140","author":"Soncini","year":"2014","journal-title":"J. Chem. Phys."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"041902","DOI":"10.1063\/5.0139706","article-title":"A nonvanishing spectral gap for AKLT models on generalized decorated graphs","volume":"64","author":"Lucia","year":"2023","journal-title":"J. Math. Phys."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"7","DOI":"10.1017\/S0370164600006635","article-title":"On self-conjugate permutations","volume":"17","author":"Muir","year":"1889","journal-title":"Proc. R. Soc. Edinb."},{"key":"ref_43","unstructured":"Muir, T. (1960). A Treatise on the Theory of Determinants, Dover."},{"key":"ref_44","unstructured":"Skiena, S. (1990). Implementary Discrete Mathematics: Combinatorics and Graph Theory with Mathematics, Addison-Wesley. Chapter 1.4.1."},{"key":"ref_45","first-page":"37","article-title":"Spectra of graphs formed by some unary operations","volume":"19","year":"1975","journal-title":"Publ. Inst. Math."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"115","DOI":"10.1090\/S0002-9947-99-02497-6","article-title":"The spectrum of infinite regular line graphs","volume":"352","author":"Shirai","year":"2000","journal-title":"Trans. Am. Math. Soc."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/j.jmaa.2008.08.036","article-title":"Clique-inserted-graphs and spectral dynamics of clique-inserting","volume":"349","author":"Zhang","year":"2009","journal-title":"J. Math. Anal. Appl."},{"key":"ref_48","first-page":"302","article-title":"A comparison on metric dimension of graphs, line graphs, and line graphs of the subdivision graphs","volume":"5","author":"Klein","year":"2012","journal-title":"Eur. J. Pure Appl. Math."},{"key":"ref_49","doi-asserted-by":"crossref","first-page":"1608","DOI":"10.1007\/s10910-013-0168-1","article-title":"Eigenvalues of saturated hydrocarbons","volume":"51","author":"Klein","year":"2013","journal-title":"J. Math. Chem."},{"key":"ref_50","first-page":"2435","article-title":"Study of the para-line graphs of certain polyphenyl chains using topological indices","volume":"8","author":"Zhang","year":"2017","journal-title":"Int. J. Biochem. Biotechnol."},{"key":"ref_51","first-page":"93","article-title":"On the para-line graphs of certain nanostructures based on typological indices","volume":"79","author":"Gao","year":"2017","journal-title":"UPB Sci. Bull. Ser. B"},{"key":"ref_52","first-page":"9915","article-title":"Topological properties of para-line graph of some convex polytopes using neighborhood M-polynomial","volume":"11","author":"Mondal","year":"2021","journal-title":"Biointerface Res. Appl. Chem."},{"key":"ref_53","first-page":"12","article-title":"On r-dynamic coloring of para-line graph of some standard graphs","volume":"10","author":"Nandini","year":"2021","journal-title":"Palest. J. Math."},{"key":"ref_54","doi-asserted-by":"crossref","first-page":"1551","DOI":"10.1007\/s10910-021-01254-2","article-title":"The spectrum of the vertex quadrangulation of a 4-regular toroidal graph and beyond","volume":"59","author":"Rosenfeld","year":"2021","journal-title":"J. Math. Chem."},{"key":"ref_55","doi-asserted-by":"crossref","unstructured":"Rosenfeld, V.R. (2024). Eigenvalue \u22121 of the vertex quadrangulation of a 4-regular graph. Axioms, 13.","DOI":"10.3390\/axioms13010072"},{"key":"ref_56","doi-asserted-by":"crossref","first-page":"C9","DOI":"10.1016\/0022-328X(87)80030-X","article-title":"The structure of polymeric dialkyl [R2SnO] (R=Me,Bu) as probed by high-resolution solid-state 119Sn NMR","volume":"331","author":"Harris","year":"1987","journal-title":"J. Organomet. Chem."},{"key":"ref_57","doi-asserted-by":"crossref","first-page":"085106","DOI":"10.1103\/PhysRevB.82.085106","article-title":"Topological phases and phase transitions on the square-octagon lattice","volume":"12","author":"Kargarian","year":"2010","journal-title":"Phys. Rev. B"},{"key":"ref_58","doi-asserted-by":"crossref","first-page":"205402","DOI":"10.1103\/PhysRevB.89.205402","article-title":"Gapless MoS2 allotrope possessing both massless Dirac and heavy fermions","volume":"89","author":"Li","year":"2014","journal-title":"Phys. Rev. B"},{"key":"ref_59","doi-asserted-by":"crossref","first-page":"6918","DOI":"10.1038\/srep06918","article-title":"Quantum magnetic phase transition in square-octagon lattice","volume":"4","author":"Bao","year":"2014","journal-title":"Sci. Rep."},{"key":"ref_60","doi-asserted-by":"crossref","first-page":"165421","DOI":"10.1103\/PhysRevB.92.165421","article-title":"Graphene-like Dirac states and quantum spin Hall insulators in square-octagonal MX2 (M = Mo,W;X = S, Se, Te) isomers","volume":"92","author":"Sun","year":"2015","journal-title":"Phys. Rev. B"},{"key":"ref_61","doi-asserted-by":"crossref","first-page":"109","DOI":"10.1016\/j.commatsci.2015.08.008","article-title":"Two-dimensional octagon-structure monolayer of nitrogen group elements and the related nano-structures","volume":"110","author":"Zhang","year":"2015","journal-title":"Comput. Mat. Sci."},{"key":"ref_62","doi-asserted-by":"crossref","first-page":"1674","DOI":"10.1038\/s41598-018-19496-7","article-title":"Electronic structure and band gap engineering of two-dimensional octagon-nitrogen","volume":"8","author":"Lin","year":"2018","journal-title":"Sci. Rep."},{"key":"ref_63","doi-asserted-by":"crossref","first-page":"30","DOI":"10.1016\/j.chphma.2022.04.009","article-title":"Two-dimensional Dirac materials: Tight-binding lattice models and material candidates","volume":"2","author":"Fan","year":"2023","journal-title":"ChemPhysMater"},{"key":"ref_64","unstructured":"Delgado-Friedrichs, O. (2023, December 22). Analyzing Periodic Nets via the Barycentre Construction, in Lecture on the Inner Workings of Systre Held in Santa Barbara in August 2008. File Systre-Lecture. Available online: http:\/\/gavrog.org\/systre-lecture.pdf."},{"key":"ref_65","doi-asserted-by":"crossref","first-page":"033071","DOI":"10.1103\/PhysRevResearch.5.033071","article-title":"Benchmarking variational quantum eigensolvers for the square-octagon-lattice Kitaev model","volume":"5","author":"Li","year":"2023","journal-title":"Phys. Rev. Res."},{"key":"ref_66","doi-asserted-by":"crossref","first-page":"114294","DOI":"10.1016\/j.matdes.2025.114294","article-title":"Unveiling the multifaceted properties of square-octagon lattices using the Hubbard model","volume":"256","author":"Abdi","year":"2025","journal-title":"Mater. Des."},{"key":"ref_67","doi-asserted-by":"crossref","first-page":"10606","DOI":"10.1039\/D5CP00722D","article-title":"Exploring the electronic properties and quantum capacitance of the square-octagon lattice for advanced electronic and energy storage applications","volume":"27","author":"Norian","year":"2025","journal-title":"Phys. Chem. Chem. Phys."},{"key":"ref_68","doi-asserted-by":"crossref","unstructured":"Vijayvargia, A., Day-Roberts, E., Botana, A.S., and Erten, O. (2025). Altermagnets with topological order in Kitaev bilayers. arXiv.","DOI":"10.1103\/km2j-3zy2"},{"key":"ref_69","unstructured":"(2025, October 03). Wikipedia, Organotin Chemistry. Available online: https:\/\/en.wikipedia.org\/wiki\/Wiki.organotin\/_chemistry."},{"key":"ref_70","doi-asserted-by":"crossref","first-page":"291","DOI":"10.1016\/j.jctb.2019.01.002","article-title":"Symmetry properties of generalized graph truncations","volume":"137","author":"Eibena","year":"2019","journal-title":"J. Comb. Theory Ser. B"},{"key":"ref_71","unstructured":"Lek\u0161e, M., and Toledo, M. (2025). Cubic vertex-transitive graphs of girth seven. arXiv."},{"key":"ref_72","doi-asserted-by":"crossref","first-page":"2612","DOI":"10.1016\/j.disc.2011.10.021","article-title":"Recursive constructions of small regular graphs of given degree and girth","volume":"312","author":"Exoo","year":"2012","journal-title":"Discrete Math."},{"key":"ref_73","doi-asserted-by":"crossref","first-page":"209","DOI":"10.1007\/s10801-012-0400-2","article-title":"Cayley cages","volume":"38","author":"Exoo","year":"2013","journal-title":"J. Algebr. Comb."},{"key":"ref_74","doi-asserted-by":"crossref","first-page":"P1.77","DOI":"10.37236\/4680","article-title":"Generalized cages","volume":"22","author":"Boben","year":"2015","journal-title":"Electron. J. Combin."},{"key":"ref_75","unstructured":"Alspach, B., and Connor, J.B. (2020). Some graph theoretical aspects of generalized truncations. arXiv."},{"key":"ref_76","doi-asserted-by":"crossref","first-page":"325","DOI":"10.26493\/1855-3974.2122.1e2","article-title":"On generalized truncations of complete graphs","volume":"19","author":"Wang","year":"2020","journal-title":"Ars Math. Contemp."},{"key":"ref_77","doi-asserted-by":"crossref","unstructured":"Dobson, T., Hujdurovi\u0107, A., Imrich, W., and Ortner, R. (2025). On cubic vertex-transitive graphs of given girth. arXiv.","DOI":"10.26493\/2590-9770.1724.18a"},{"key":"ref_78","doi-asserted-by":"crossref","first-page":"1014","DOI":"10.1007\/s10910-016-0716-6","article-title":"The truncation of a cage graph","volume":"55","author":"Diudea","year":"2017","journal-title":"J. Math. Chem."},{"key":"ref_79","doi-asserted-by":"crossref","first-page":"215","DOI":"10.26493\/1855-3974.665.4b6","article-title":"On automorphism groups of graph truncations","volume":"8","author":"Alspach","year":"2015","journal-title":"Ars Math. Contemp."},{"key":"ref_80","doi-asserted-by":"crossref","first-page":"259","DOI":"10.1016\/S0024-3795(97)83595-1","article-title":"Compact graphs and equitable partitions","volume":"255","author":"Godsil","year":"1997","journal-title":"Linear Algebra Appl."},{"key":"ref_81","doi-asserted-by":"crossref","first-page":"121","DOI":"10.1016\/S0024-3795(02)00434-2","article-title":"Equitable switching and spectra of graphs","volume":"359","author":"Teranishi","year":"2003","journal-title":"Linear Algebra Appl."},{"key":"ref_82","first-page":"1029","article-title":"Equitable block colouring for systems of 4-kites","volume":"11","author":"Bonacini","year":"2017","journal-title":"Appl. Math. Sci."},{"key":"ref_83","doi-asserted-by":"crossref","first-page":"432","DOI":"10.1016\/j.laa.2017.06.045","article-title":"Extensions and applications of equitable decompositions for graphs with symmetries","volume":"532","author":"Francis","year":"2017","journal-title":"Linear Algebra Appl."},{"key":"ref_84","doi-asserted-by":"crossref","first-page":"25","DOI":"10.1016\/j.automatica.2017.01.018","article-title":"Almost equitable partitions and new necessary conditions for network controllability","volume":"80","author":"Aguilar","year":"2017","journal-title":"Automatica"},{"key":"ref_85","doi-asserted-by":"crossref","first-page":"07315","DOI":"10.1063\/1.4997385","article-title":"Synchronization and equitable partitions in weighted networks","volume":"28","author":"Aguiar","year":"2018","journal-title":"Chaos"},{"key":"ref_86","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1016\/j.laa.2019.04.013","article-title":"On the spectrum of an equitable quotient matrix and its application","volume":"577","author":"You","year":"2019","journal-title":"Linear Algebra Appl."},{"key":"ref_87","doi-asserted-by":"crossref","first-page":"112039","DOI":"10.1016\/j.disc.2020.112039","article-title":"Equitable 2-partitions of the Hamming graphs with the second eigenvalue","volume":"343","author":"Mogilnykh","year":"2020","journal-title":"Discrete Math."},{"key":"ref_88","doi-asserted-by":"crossref","first-page":"112351","DOI":"10.1016\/j.disc.2021.112351","article-title":"Equitable partition of planar graphs","volume":"344","author":"Kim","year":"2021","journal-title":"Discrete Math."},{"key":"ref_89","first-page":"73","article-title":"Applications of a theorem on partitioned matrices","volume":"62 B","author":"Haynsworth","year":"1959","journal-title":"J. Res. Nat. Bureau Stand."},{"key":"ref_90","first-page":"123","article-title":"\u00dcber Spectrum, Automorphismengruppe und Teiler eines Graphen","volume":"15","author":"Petersdorf","year":"1969","journal-title":"Wiss. Z. Tech. Hochsch. Ilmenau"},{"key":"ref_91","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/0024-3795(87)90064-4","article-title":"Divisors and the spectrum of infinite graphs","volume":"91","author":"Mohar","year":"1987","journal-title":"Linear Algebra Appl."},{"key":"ref_92","doi-asserted-by":"crossref","first-page":"1941","DOI":"10.1007\/s10910-017-0773-5","article-title":"The toroidal unit cell of a quasicrystal","volume":"55","author":"Rosenfeld","year":"2017","journal-title":"J. Math. Chem."},{"key":"ref_93","doi-asserted-by":"crossref","unstructured":"Mizutani, U. (2001). Introduction to the Electron Theory of Metals, Cambridge University Press.","DOI":"10.1017\/CBO9780511612626"},{"key":"ref_94","doi-asserted-by":"crossref","unstructured":"Hell, P., and Ne\u0161et\u0159il, J. (2004). Graphs and Homomorphisms, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780198528173.001.0001"},{"key":"ref_95","first-page":"32","article-title":"The main part of the spectrum, divisors and switching of graphs","volume":"23","year":"1978","journal-title":"Publ. Inst. Math."},{"key":"ref_96","doi-asserted-by":"crossref","first-page":"638","DOI":"10.1021\/ci9900231","article-title":"A group-theoretical bound for the number of main eigenvalues of a graph","volume":"39","author":"Fowler","year":"1999","journal-title":"J. Chem. Inf. Comput. Sci."},{"key":"ref_97","doi-asserted-by":"crossref","first-page":"1758","DOI":"10.1007\/s10910-016-0653-4","article-title":"Semigroup theory of symmetry","volume":"54","author":"Rosenfeld","year":"2016","journal-title":"J. Math. Chem"},{"key":"ref_98","first-page":"203","article-title":"Endomorphisms of a weighted molecular graph and its spectrum","volume":"40","author":"Rosenfeld","year":"1999","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_99","first-page":"313","article-title":"Rainbow graphs","volume":"Volume 10","author":"Arasu","year":"2002","journal-title":"Codes and Designs. Proceedings of the Conference Honoring Professor Dijen K. Ray-Chaudhuri on the Occasion of His 65th Birthday, the Ohio State University, Columbus, OH, USA, 18\u201321 May 2000"},{"key":"ref_100","doi-asserted-by":"crossref","unstructured":"Lov\u00e1sz, L., and Plummer, M.D. (2009). Matching Theory, American Mathematical Society.","DOI":"10.1090\/chel\/367"},{"key":"ref_101","doi-asserted-by":"crossref","first-page":"361","DOI":"10.1016\/0022-2860(82)85267-8","article-title":"Topological effects on MO energies","volume":"84","author":"Polansky","year":"1982","journal-title":"J. Mol. Struct."},{"key":"ref_102","doi-asserted-by":"crossref","first-page":"196","DOI":"10.1515\/zna-1983-0216","article-title":"Topological effects on MO energies, II","volume":"38a","author":"Polansky","year":"1983","journal-title":"Z. Naturforsch."},{"key":"ref_103","doi-asserted-by":"crossref","first-page":"916","DOI":"10.1515\/zna-1983-0817","article-title":"Topological effect on MO energies. III. Heterocyclic systems with non-isomorphic partial structures","volume":"38a","author":"Fabian","year":"1983","journal-title":"Z. Naturforsch."},{"key":"ref_104","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/BF00798416","article-title":"Topological effect on MO energies, IV. The total \u03c0-electron energy of S- and T-isomers","volume":"115","author":"Graovac","year":"1984","journal-title":"Monatsch. Chem."},{"key":"ref_105","doi-asserted-by":"crossref","first-page":"281","DOI":"10.1016\/0022-2860(84)80152-0","article-title":"Topological effects displayed in absorption and photoelectron spectra","volume":"113","author":"Polansky","year":"1984","journal-title":"J. Mol. Struct."},{"key":"ref_106","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/BF00547896","article-title":"The effect of molecular topology on \u03c0-molecular-orbital energies","volume":"67","author":"Motoc","year":"1985","journal-title":"Theoret. Chim. Acta"},{"key":"ref_107","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/0166-218X(88)90014-5","article-title":"Spectral properties of some structurally related graphs","volume":"19","author":"Gutman","year":"1988","journal-title":"Discrete Appl. Math."},{"key":"ref_108","first-page":"277","article-title":"Topological properties of some novel S,T-isomers (I)","volume":"28","author":"Elken","year":"1992","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_109","first-page":"179","article-title":"Topological properties of some novel S,T-isomers II","volume":"30","author":"Li","year":"1994","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_110","doi-asserted-by":"crossref","first-page":"323","DOI":"10.1515\/spma-2018-0027","article-title":"Spectra of graphs resulting from various graph operations and products: A survey","volume":"6","author":"Barik","year":"2018","journal-title":"Spec. Matrices"},{"key":"ref_111","doi-asserted-by":"crossref","first-page":"119","DOI":"10.1016\/S0012-365X(98)00069-7","article-title":"Factoring cardinal product graphs in polynomial time","volume":"192","author":"Imrich","year":"1998","journal-title":"Discrete Math."},{"key":"ref_112","doi-asserted-by":"crossref","unstructured":"Hammack, R., Imrich, W., and Klav\u017ear, S. (2011). Handbook of Product Graphs, CRC Press Taylor and Francis Group A Chapman & Hall Book.","DOI":"10.1201\/b10959"},{"key":"ref_113","unstructured":"(2024, December 15). Wikipedia, Tensor Product of Graphs. Available online: https:\/\/en.wikipedia.org\/wiki\/Tensor_product_of_graphs."},{"key":"ref_114","doi-asserted-by":"crossref","unstructured":"Horn, R.A., and Johnson, C.R. (1991). Topics in Matrix Analysis, Cambridge University Press.","DOI":"10.1017\/CBO9780511840371"},{"key":"ref_115","unstructured":"(2025, October 09). Wikipedia, Kronecker Product. Available online: https:\/\/en.wikipedia.org\/wiki\/Kronecker_product."},{"key":"ref_116","doi-asserted-by":"crossref","first-page":"1850","DOI":"10.1007\/s10910-019-01042-z","article-title":"Looking into the future of molecules with novel topological symmetries","volume":"57","author":"Rosenfeld","year":"2019","journal-title":"J. Math. Chem."},{"key":"ref_117","doi-asserted-by":"crossref","unstructured":"Imrich, W., Klav\u017ear, S., and Rall, D.F. (2008). Topics in Graph Theory: Graphs and Their Cartesian Products, A. K. Peters Ltd.","DOI":"10.1201\/b10613"},{"key":"ref_118","unstructured":"(2025, October 19). Wikipedia, Cartesian Product of Graphs. Available online: https:\/\/en.wikipedia.org\/wiki\/Cartesian_product_of_graphs."},{"key":"ref_119","doi-asserted-by":"crossref","unstructured":"Bapat, R.B. (2014). Graphs and Matrices, Hindustan Book Agency.","DOI":"10.1007\/978-1-4471-6569-9"},{"key":"ref_120","doi-asserted-by":"crossref","first-page":"515","DOI":"10.4153\/CJM-1957-060-7","article-title":"Graphs with given group and given graph-theoretical properties","volume":"9","author":"Sabidussi","year":"1957","journal-title":"Canad. J. Math."},{"key":"ref_121","first-page":"365","article-title":"Graph theory and Molecular orbitals. II. Croat","volume":"44","author":"Gutman","year":"1972","journal-title":"Chem. Acta"},{"key":"ref_122","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/0022-2860(75)80099-8","article-title":"Graphical studies on the relations between the structure and reactivity of conjugated systems: The role of non-bonding molecular orbitals","volume":"28","author":"Gutman","year":"1975","journal-title":"J. Mol. Struct."},{"key":"ref_123","doi-asserted-by":"crossref","unstructured":"Graovac, A., Gutman, I., and Trinajsti\u0107, N. (1977). Topological Approach to the Chemistry of Conjugated Molecules (Lecture Notes in Chemistry, 4), Springer.","DOI":"10.1007\/978-3-642-93069-0_2"},{"key":"ref_124","doi-asserted-by":"crossref","unstructured":"Gutman, I., and Polansky, O.E. (1985). Mathematical Concepts in Organic Chemistry, Springer.","DOI":"10.1515\/9783112570180"},{"key":"ref_125","unstructured":"Sciriha, I., and Farrugia, A. (2021). From Nut Graphs to the Molecular Structure and Conductivity, Faculty of Science, University of Kragujevac."},{"key":"ref_126","doi-asserted-by":"crossref","first-page":"1043","DOI":"10.1080\/00268979500101651","article-title":"Structural origin of specific eigenvalues in chemical graphs of planar molecules. Molecular orbital functional groups","volume":"85","author":"Dias","year":"1995","journal-title":"Mol. Phys."},{"key":"ref_127","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1016\/0895-7177(88)90571-7","article-title":"On non-symmetry equivalence","volume":"11","author":"King","year":"1988","journal-title":"Math. Comput. Model."},{"key":"ref_128","doi-asserted-by":"crossref","unstructured":"Cvetkovi\u0107, D., Rowlinson, P., and Simi\u0107, S. (1997). Eigenspaces of Graphs, Cambridge University Press.","DOI":"10.1017\/CBO9781139086547"},{"key":"ref_129","doi-asserted-by":"crossref","unstructured":"Cvetkovi\u0107, D., Rowlinson, P., and Simi\u0107, S. (2004). Spectral Generalizations of Line Graphs. On Graphs with Least Eigenvalue \u22122, Cambridge University Press.","DOI":"10.1017\/CBO9780511751752"},{"key":"ref_130","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1002\/hlca.19530360125","article-title":"Das Kompositions-Prinzip: Eine anshchauliche Methode zur elektronen theoretischen Behandlung nicht oder niedrig symmetrishcher Molekeln im Rahmen der MO-Theorie","volume":"36","author":"Heilbronner","year":"1953","journal-title":"Helv. Chim. Acta"},{"key":"ref_131","unstructured":"Marcus, M., and Minc, H. (1964). A Survey of Matrix Theory and Matrix Inequalities, Allyn and Bacon, Inc."},{"key":"ref_132","first-page":"113","article-title":"Interlacing results on matrices associated with graphs","volume":"68","author":"Hall","year":"2009","journal-title":"J. Combin. Math. Combin. Comput. (JCMCC)"},{"key":"ref_133","unstructured":"Horn, R.A., and Johnson, C.R. (2013). Matrix Analysis, Cambridge University Press. [2nd ed.]."},{"key":"ref_134","first-page":"899","article-title":"On the characteristic polynomial of t-tuple and b-bridges coalescence graphs","volume":"42","author":"Ibrahim","year":"2018","journal-title":"Southeast Asian Bull. Math."},{"key":"ref_135","doi-asserted-by":"crossref","first-page":"1453","DOI":"10.1039\/F29747001453","article-title":"Graphical method for factorizing secular determinants of H\u00fcckel molecular orbital theory","volume":"70","author":"McClelland","year":"1974","journal-title":"J. Chem. Soc. Faraday Trans. II"},{"key":"ref_136","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1007\/BF00555691","article-title":"Eigenvalues of graphs with threefold symmetry","volume":"53","year":"1979","journal-title":"Theor. Chim. Acta"},{"key":"ref_137","doi-asserted-by":"crossref","first-page":"1363","DOI":"10.1080\/00268977900101001","article-title":"Eigenvalues of graphs with twofold symmetry","volume":"37","year":"1979","journal-title":"Mol. Phys."},{"key":"ref_138","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1007\/BF00551119","article-title":"Spectral analysis of graphs by cyclic automorphism groups","volume":"58","author":"Davidson","year":"1981","journal-title":"Theor. Chim. Acta"},{"key":"ref_139","doi-asserted-by":"crossref","first-page":"211","DOI":"10.1016\/S0065-3276(08)60294-4","article-title":"Symmetry rules in the graph theory of molecular orbitals","volume":"13","author":"Yan","year":"1981","journal-title":"Adv. Quantum Chem."},{"key":"ref_140","doi-asserted-by":"crossref","first-page":"911","DOI":"10.1039\/f29827800911","article-title":"Eigenvalues of the topological matrix, splitting of graphs with symmetrical components and alternant graphs","volume":"78","author":"McClelland","year":"1982","journal-title":"J. Chem. Soc. Faraday Trans. II"},{"key":"ref_141","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1080\/00268978200100151","article-title":"On the graphical factorization of H\u00fcckel characteristic equations","volume":"45","author":"McClelland","year":"1982","journal-title":"Mol. Phys."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/12\/907\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,12]],"date-time":"2025-12-12T05:12:28Z","timestamp":1765516348000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/12\/907"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,10]]},"references-count":141,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["axioms14120907"],"URL":"https:\/\/doi.org\/10.3390\/axioms14120907","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2025,12,10]]}}}