{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,29]],"date-time":"2025-12-29T11:08:40Z","timestamp":1767006520103,"version":"build-2065373602"},"reference-count":30,"publisher":"MDPI AG","issue":"7","license":[{"start":{"date-parts":[[2018,7,5]],"date-time":"2018-07-05T00:00:00Z","timestamp":1530748800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Chemical graph theory plays an important role in modeling and designing any chemical structure. The molecular topological descriptors are the numerical invariants of a molecular graph and are very useful for predicting their bioactivity. In this paper, we study the chemical graph of the crystal structure of titanium difluoride TiF2 and the crystallographic structure of cuprite Cu2O. Furthermore, we compute degree-based topological indices, mainly ABC, GA, ABC4, GA5 and general Randi\u0107 indices. Furthermore, we also give exact results of these indices for the crystal structure of titanium difluoride TiF2 and the crystallographic structure of cuprite Cu2O.<\/jats:p>","DOI":"10.3390\/sym10070265","type":"journal-article","created":{"date-parts":[[2018,7,5]],"date-time":"2018-07-05T10:52:44Z","timestamp":1530787964000},"page":"265","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":17,"title":["Topological Properties of Crystallographic Structure of Molecules"],"prefix":"10.3390","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9620-7692","authenticated-orcid":false,"given":"Jia-Bao","family":"Liu","sequence":"first","affiliation":[{"name":"School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-2607-4847","authenticated-orcid":false,"given":"Muhammad Kamran","family":"Siddiqui","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus, Sahiwal 57000, Pakistan"}]},{"given":"Manzoor Ahmad","family":"Zahid","sequence":"additional","affiliation":[{"name":"Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus, Sahiwal 57000, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-8132-1580","authenticated-orcid":false,"given":"Muhammad","family":"Naeem","sequence":"additional","affiliation":[{"name":"Department of Mathematics, The University of Lahore, Pakpattan Campus, Pakpattan 57400, Pakistan"}]},{"given":"Abdul Qudair","family":"Baig","sequence":"additional","affiliation":[{"name":"Department of Mathematics, The University of Lahore, Pakpattan Campus, Pakpattan 57400, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2018,7,5]]},"reference":[{"key":"ref_1","first-page":"429","article-title":"Computing fourth atom bond connectivity index of V-phenylenic nanotubes and nanotori","volume":"60","author":"Farahani","year":"2013","journal-title":"Acta Chim. Slov."},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Imran, M., Ali, M.A., Ahmad, S., Siddiqui, M.K., and Baig, A.Q. (2018). Topological sharacterization of the symmetrical structure of bismuth tri-iodide. Symmetry, 10.","DOI":"10.3390\/sym10060201"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"6540754","DOI":"10.1155\/2017\/6540754","article-title":"Molecular descriptors of nanotube, oxide, silicate, and triangulene networks","volume":"2017","author":"Gao","year":"2017","journal-title":"J. Chem."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Wu, W., Zhang, C., Lin, W., Chen, Q., Guo, X., and Qian, Y. (2015). Quantitative Structure-Property Relationship (QSPR) Modeling of Drug-Loaded Polymeric Micelles via Genetic Function Approximation. PLoS ONE, 10.","DOI":"10.1371\/journal.pone.0119575"},{"key":"ref_5","first-page":"144","article-title":"Computing topological indices of honeycomb derived networks","volume":"18","author":"Hayat","year":"2015","journal-title":"Roman. J. Inf. Sci. Technol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"303","DOI":"10.17344\/acsi.2017.3925","article-title":"Possibility of C38 and Si19Ge19 nanocages in anode of metal ion batteries: computational examination","volume":"65","author":"Bie","year":"2018","journal-title":"Acta Chim. Slov."},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Holgate, S.A. (2009). Understanding Solid State Physics, CRC Press.","DOI":"10.1201\/b15977"},{"key":"ref_8","unstructured":"Van, Z.B. (2017). Section 2.3: Energy Bands. Principles of Semiconductor Devices. [Ph.D Thesis, Electrical, Computer, Energy Engineering Department, University of Colorado]."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"132","DOI":"10.1016\/j.amc.2016.01.041","article-title":"On Zagreb indices, Zagreb polynomials of some nanostar dendrimers","volume":"280","author":"Siddiqui","year":"2016","journal-title":"Appl. Math. Comput."},{"key":"ref_10","first-page":"884","article-title":"Computing topological indicesof certain networks","volume":"18","author":"Siddiqui","year":"2016","journal-title":"J. Optoelectron. Adv. Mater."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"6609","DOI":"10.1021\/ja00856a001","article-title":"On characterization of molecular branching","volume":"97","year":"1975","journal-title":"J. Am. Chem. Soc."},{"key":"ref_12","first-page":"225","article-title":"Graphs of extremal weights","volume":"50","year":"1998","journal-title":"Ars Comb."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"819","DOI":"10.1021\/ci980039b","article-title":"The vertex-connectivity index revisited","volume":"38","author":"Amic","year":"1998","journal-title":"J. Chem. Inf. Comput. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Shao, Z., Siddiqui, M.K., and Muhammad, M.H. (2018). Computing zagreb indices and zagreb polynomials for symmetrical nanotubes. Symmetry, 10.","DOI":"10.3390\/sym10070244"},{"key":"ref_15","unstructured":"Li, X., and Gutman, I. (2006). Mathematical Aspects of Randi\u0107 type Molecular Structure Descriptors, Mathematical Chemistry Monographs No. 1."},{"key":"ref_16","first-page":"849","article-title":"An atom-bond connectivity inde. Modelling the enthalpy of formation of alkanes","volume":"37A","author":"Estrada","year":"1998","journal-title":"Indian J. Chem."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1369","DOI":"10.1007\/s10910-009-9520-x","article-title":"Topological index based on the ratios of geometrical and arithmetical means of end-vertex degrees of edges","volume":"46","author":"Furtula","year":"2009","journal-title":"J. Math. Chem."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/0009-2614(72)85099-1","article-title":"Graph theory and molecular orbitals, Total \u03c0-electron energy of alternant hydrocarbons","volume":"17","author":"Gutman","year":"1972","journal-title":"Chem. Phys. Lett."},{"key":"ref_19","first-page":"83","article-title":"The first Zagreb index 30 years after","volume":"50","author":"Gutman","year":"2004","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_20","first-page":"1419","article-title":"Computing ABC4 index of nanostar dendrimers","volume":"4","author":"Ghorbani","year":"2010","journal-title":"Optoelectron. Adv. Mater.-Rapid Commun."},{"key":"ref_21","first-page":"33","article-title":"Computing fifth geometric\u2013arithmetic index for nanostar dendrimers","volume":"1","author":"Graovac","year":"2011","journal-title":"J. Math. Nanosci."},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Gao, W., Siddiqui, M.K., Naeem, M., and Rehman, N.A. (2017). Topological Characterization of Carbon Graphite and Crystal Cubic Carbon Structures. Molecules, 22.","DOI":"10.3390\/molecules22091496"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"27604","DOI":"10.1109\/ACCESS.2018.2831910","article-title":"On the maximum ABC index of graphs with prescribed size and without pendent vertices","volume":"6","author":"Shao","year":"2018","journal-title":"IEEE Access"},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"298","DOI":"10.1016\/j.amc.2017.07.075","article-title":"On the maximum ABC index of graphs without pendent vertices","volume":"315","author":"Shao","year":"2017","journal-title":"Appl. Math. Comput."},{"key":"ref_25","doi-asserted-by":"crossref","unstructured":"Imran, M., Siddiqui, M.K., Naeem, M., and Iqbal, M.A. (2018). On Topological Properties of Symmetric Chemical Structures. Symmetry, 10.","DOI":"10.3390\/sym10050173"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"258","DOI":"10.1016\/j.jsps.2016.04.012","article-title":"Forgotten Topological Index of Chemical Structure in Drugs","volume":"24","author":"Gao","year":"2016","journal-title":"Saudi Pharm. J."},{"key":"ref_27","first-page":"52","article-title":"Polymorphic crystallization of Cu2O compound","volume":"16","author":"Chen","year":"2014","journal-title":"CrystEngComm"},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3756","DOI":"10.1021\/ja8095575","article-title":"Nanowire-Based All-Oxide Solar Cells","volume":"131","author":"Yuhas","year":"2009","journal-title":"J. Am. Chem. Soc."},{"key":"ref_29","doi-asserted-by":"crossref","first-page":"867","DOI":"10.1021\/cm052256f","article-title":"Nearly Monodisperse Cu2O and CuO Nanospheres: Preparation and Applications for Sensitive Gas Sensors","volume":"18","author":"Zhang","year":"2006","journal-title":"Chem. Mater."},{"key":"ref_30","unstructured":"Cotton, F.A., Wilkinson, G., Murillo, C.A., and Bochmann, M. (1999). Advanced Inorganic Chemistry, John Wiley and Sons."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/7\/265\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T15:11:28Z","timestamp":1760195488000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/10\/7\/265"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,7,5]]},"references-count":30,"journal-issue":{"issue":"7","published-online":{"date-parts":[[2018,7]]}},"alternative-id":["sym10070265"],"URL":"https:\/\/doi.org\/10.3390\/sym10070265","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2018,7,5]]}}}