{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,19]],"date-time":"2025-12-19T09:49:03Z","timestamp":1766137743437,"version":"build-2065373602"},"reference-count":41,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,7,29]],"date-time":"2021-07-29T00:00:00Z","timestamp":1627516800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100011033","name":"Agencia Estatal de Investigaci\u00f3n","doi-asserted-by":"publisher","award":["PID2019-106433GBI00\/AEI\/10.13039\/501100011033"],"award-info":[{"award-number":["PID2019-106433GBI00\/AEI\/10.13039\/501100011033"]}],"id":[{"id":"10.13039\/501100011033","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Madrid Government (Comunidad de Madrid-Spain)","award":["Multiannual Agreement with UC3M"],"award-info":[{"award-number":["Multiannual Agreement with UC3M"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>We perform a detailed computational study of the recently introduced Sombor indices on random networks. Specifically, we apply Sombor indices on three models of random networks: Erd\u00f6s-R\u00e9nyi networks, random geometric graphs, and bipartite random networks. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the network, scale with the average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random networks and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the adjacency matrix.<\/jats:p>","DOI":"10.3390\/e23080976","type":"journal-article","created":{"date-parts":[[2021,7,29]],"date-time":"2021-07-29T21:21:21Z","timestamp":1627593681000},"page":"976","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":13,"title":["Normalized Sombor Indices as Complexity Measures of Random Networks"],"prefix":"10.3390","volume":"23","author":[{"given":"R.","family":"Aguilar-S\u00e1nchez","sequence":"first","affiliation":[{"name":"Facultad de Ciencias Qu\u00edmicas, Benem\u00e9rita Universidad Aut\u00f3noma de Puebla, Puebla 72570, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1748-9901","authenticated-orcid":false,"given":"J. A.","family":"M\u00e9ndez-Berm\u00fadez","sequence":"additional","affiliation":[{"name":"Instituto de F\u00edsica, Benem\u00e9rita Universidad Aut\u00f3noma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2851-7442","authenticated-orcid":false,"given":"Jos\u00e9 M.","family":"Rodr\u00edguez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1ticas, Universidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Legan\u00e9s, Madrid, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4352-5109","authenticated-orcid":false,"given":"Jos\u00e9 M.","family":"Sigarreta","sequence":"additional","affiliation":[{"name":"Facultad de Matem\u00e1ticas, Universidad Aut\u00f3noma de Guerrero, Carlos E. Adame No. 54 Col. Garita, Acapulco 39650, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,29]]},"reference":[{"key":"ref_1","first-page":"11","article-title":"Geometric approach to degree-based topological indices: Sombor indices","volume":"86","author":"Gutman","year":"2021","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1","DOI":"10.14445\/23939133\/IJAC-V8I1P101","article-title":"Computation of Sombor indices of certain networks","volume":"8","author":"Kulli","year":"2021","journal-title":"SSRG Int. J. Appl. Chem."},{"key":"ref_3","unstructured":"Lina, Z., Zhoub, T., Kullic, V.R., and Miao, L. (2021). On the first Banhatti-Sombor index. arXiv."},{"key":"ref_4","first-page":"11","article-title":"On the Sombor index of graphs","volume":"3","author":"Reti","year":"2021","journal-title":"Contrib. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"115","DOI":"10.14445\/22315373\/IJMTT-V65I11P512","article-title":"The (a,b)-KA indices of polycyclic aromatic hydrocarbons and benzenoid systems","volume":"65","author":"Kulli","year":"2019","journal-title":"Int. J. Math. Trends Technol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1007\/s10910-009-9542-4","article-title":"On general sum-connectivity index","volume":"47","author":"Zhou","year":"2010","journal-title":"J. Math. Chem."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"37","DOI":"10.22457\/apam.v23n1a06812","article-title":"\u03b4-Sombor index and its exponential for certain nanotubes","volume":"23","author":"Kulli","year":"2021","journal-title":"Ann. Pure Appl. Math."},{"key":"ref_8","first-page":"126018","article-title":"Sombor index of chemical graphs","volume":"399","author":"Cruz","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1098","DOI":"10.1007\/s10910-021-01232-8","article-title":"Extremal values of the Sombor index in unicyclic and bicyclic graphs","volume":"59","author":"Cruz","year":"2021","journal-title":"J. Math. Chem."},{"key":"ref_10","unstructured":"Ghanbari, N., and Alikhani, S. (2021). Sombor index of certain graphs. arXiv."},{"key":"ref_11","first-page":"715","article-title":"Sombor index of polymers","volume":"86","author":"Alikhani","year":"2021","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Fang, X., You, L., and Liu, H. (2021). The expected values of Sombor indices in random hexagonal chains, phenylene chains and Sombor indices of some chemical graphs. Int. J. Quantum Chem.","DOI":"10.22541\/au.161825077.77699117\/v1"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"e26622","DOI":"10.1002\/qua.26622","article-title":"Molecular trees with extremal values of Sombor indices","volume":"121","author":"Deng","year":"2021","journal-title":"Quantum Chem."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"Liu, H. (2021). Ordering chemical graphs by their Sombor indices. arXiv.","DOI":"10.46793\/match.87-1.005L"},{"key":"ref_15","unstructured":"Liu, H. (2021). Maximum Sombor index among cacti. arXiv."},{"key":"ref_16","first-page":"2022","article-title":"Ordering chemical graphs by Sombor indices and its applications","volume":"87","author":"Liu","year":"2021","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_17","unstructured":"Zhou, T., Lin, Z., and Miao, L. (2021). The Sombor index of trees and unicyclic graphs with given matching number. arXiv."},{"key":"ref_18","unstructured":"Zhou, T., Lin, Z., and Miao, L. (2021). The Sombor index of trees and unicyclic graphs with given maximum degree. arXiv."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1","DOI":"10.30538\/psrp-odam2021.0047","article-title":"Some basic properties of Sombor indices","volume":"4","author":"Gutman","year":"2021","journal-title":"Open J. Discret. Appl. Math."},{"key":"ref_20","first-page":"341","article-title":"On some mathematical properties of Sombor indices","volume":"11","author":"Milovanovic","year":"2021","journal-title":"Bull. Int. Math. Virtual Inst."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Das, K.C., Cevik, A.S., Cangul, I.N., and Shang, Y. (2021). On Sombor index. Symmetry, 13.","DOI":"10.3390\/sym13010140"},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"87","DOI":"10.1016\/j.dam.2021.04.014","article-title":"General properties on Sombor indices","volume":"299","author":"Rada","year":"2021","journal-title":"Discret. Appl. Math."},{"key":"ref_23","first-page":"5","article-title":"Some results on the Sombor indices of graphs","volume":"3","author":"Ali","year":"2021","journal-title":"Contrib. Math."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Wang, Z., Mao, Y., Li, Y., and Furtula, B. (2021). On relations between Sombor and other degree-based indices. J. Appl. Math. Comput.","DOI":"10.1007\/s12190-021-01516-x"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"445","DOI":"10.2298\/JSC201215006R","article-title":"Chemical applicability of Sombor indices","volume":"86","author":"Redzepovic","year":"2021","journal-title":"J. Serb. Chem Soc."},{"key":"ref_26","doi-asserted-by":"crossref","unstructured":"Lin, Z. (2021). On the spectral radius, energy and Estrada index of the Sombor matrix of graphs. arXiv.","DOI":"10.1142\/S1793830921501081"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"107","DOI":"10.1007\/BF02478357","article-title":"Connectivity of random nets","volume":"13","author":"Solomonoff","year":"1951","journal-title":"Bull. Math. Biophys."},{"key":"ref_28","first-page":"290","article-title":"On random graphs","volume":"6","year":"1959","journal-title":"Publ. Math. Debr. Hung."},{"key":"ref_29","first-page":"17","article-title":"On the evolution of random graphs","volume":"5","year":"1960","journal-title":"Inst. Hung. Acad. Sci."},{"key":"ref_30","first-page":"261","article-title":"On the strength of connectedness of a random graph","volume":"12","year":"1961","journal-title":"Acta Math. Hung."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"016121","DOI":"10.1103\/PhysRevE.66.016121","article-title":"Random geometric graphs","volume":"66","author":"Dall","year":"2002","journal-title":"Phys. Rev. E"},{"key":"ref_32","doi-asserted-by":"crossref","unstructured":"Penrose, M. (2003). Random Geometric Graphs, Oxford University Press.","DOI":"10.1093\/acprof:oso\/9780198506263.001.0001"},{"key":"ref_33","first-page":"125137","article-title":"Computational and analytical studies of the Randi\u0107 index in Erd\u00f6s\u2013R\u00e9nyi models","volume":"377","author":"Sigarreta","year":"2020","journal-title":"Appl. Math. Comput."},{"key":"ref_34","first-page":"395","article-title":"Computational and analytical studies of the harmonic index in Erd\u00f6s\u2013R\u00e9nyi models","volume":"85","author":"Sigarreta","year":"2021","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"042805","DOI":"10.1103\/PhysRevE.91.042805","article-title":"Random rectangular graphs","volume":"91","author":"Estrada","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"032122","DOI":"10.1103\/PhysRevE.91.032122","article-title":"Universality in the spectral and eigenfunction properties of random networks","volume":"91","author":"Rodrigues","year":"2015","journal-title":"Phys. Rev. E"},{"key":"ref_37","doi-asserted-by":"crossref","first-page":"753","DOI":"10.1093\/comnet\/cnx053","article-title":"Weighted random-geometric and random-rectangular graphs: Spectral and eigenfunction properties of the adjacency matrix","volume":"6","author":"Alonso","year":"2018","journal-title":"J. Complex Netw."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"042306","DOI":"10.1103\/PhysRevE.102.042306","article-title":"Topological versus spectral properties of random geometric graphs","volume":"102","author":"Rodrigues","year":"2020","journal-title":"Phys. Rev. E"},{"key":"ref_39","unstructured":"Metha, M.L. (2004). Random Matrices, Elsevier."},{"key":"ref_40","doi-asserted-by":"crossref","unstructured":"Haake, F. (2010). Quantum Signatures of Chaos, Springer.","DOI":"10.1007\/978-3-642-05428-0"},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"123298","DOI":"10.1016\/j.physa.2019.123298","article-title":"Normal mode analysis of spectra of random networks","volume":"545","author":"Fossion","year":"2020","journal-title":"Phys. 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