{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,2]],"date-time":"2026-06-02T20:15:36Z","timestamp":1780431336219,"version":"3.54.1"},"reference-count":32,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2020,8,11]],"date-time":"2020-08-11T00:00:00Z","timestamp":1597104000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001807","name":"Funda\u00e7\u00e3o de Amparo \u00e0 Pesquisa do Estado de S\u00e3o Paulo","doi-asserted-by":"publisher","award":["2019\/ 06931-2"],"award-info":[{"award-number":["2019\/ 06931-2"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100013704","name":"CONACyT","doi-asserted-by":"publisher","award":["2019-000009-01EXTV-00067"],"award-info":[{"award-number":["2019-000009-01EXTV-00067"]}],"id":[{"id":"10.13039\/501100013704","id-type":"DOI","asserted-by":"publisher"}]},{"name":"PRODEP-SEP","award":["511-6\/2019.-11821"],"award-info":[{"award-number":["511-6\/2019.-11821"]}]},{"name":"Ministerio de Economia y Competitividad, Agencia Estatal de Investigacion (AEI)","award":["MTM2016-78227-C2-1-P"],"award-info":[{"award-number":["MTM2016-78227-C2-1-P"]}]},{"DOI":"10.13039\/501100008530","name":"European Regional Development Fund","doi-asserted-by":"publisher","award":["MTM2017-90584-REDT"],"award-info":[{"award-number":["MTM2017-90584-REDT"]}],"id":[{"id":"10.13039\/501100008530","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We perform a detailed (computational) scaling study of well-known general indices (the first and second variable Zagreb indices, M1\u03b1(G) and M2\u03b1(G), and the general sum-connectivity index, \u03c7\u03b1(G)) as well as of general versions of indices of interest: the general inverse sum indeg index ISI\u03b1(G) and the general first geometric-arithmetic index GA\u03b1(G) (with \u03b1\u2208R). We apply these indices on two models of random networks: Erd\u00f6s\u2013R\u00e9nyi (ER) random networks GER(nER,p) and random geometric (RG) graphs GRG(nRG,r). The ER random networks are formed by nER vertices connected independently with probability p\u2208[0,1]; while the RG graphs consist of nRG vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidean distance is less or equal than the connection radius r\u2208[0,2]. Within a statistical random matrix theory approach, we show that the average values of the indices normalized to the network size scale with the average degree k of the corresponding random network models, where kER=(nER\u22121)p and kRG=(nRG\u22121)(\u03c0r2\u22128r3\/3+r4\/2). That is, X(GER)\/nER\u2248X(GRG)\/nRG if kER=kRG, with X representing any of the general indices listed above. With this work, we give a step forward in the scaling of topological indices since we have found a scaling law that covers different network models. Moreover, taking into account the symmetries of the topological indices we study here, we propose to establish their statistical analysis as a generic tool for studying average properties of random networks. In addition, we discuss the application of specific topological indices as complexity measures for random networks.<\/jats:p>","DOI":"10.3390\/sym12081341","type":"journal-article","created":{"date-parts":[[2020,8,11]],"date-time":"2020-08-11T10:56:12Z","timestamp":1597143372000},"page":"1341","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["Computational Properties of General Indices on Random Networks"],"prefix":"10.3390","volume":"12","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":[{"vocabulary":"crossref","role":"author"}]},{"given":"I. F.","family":"Herrera-Gonz\u00e1lez","sequence":"additional","affiliation":[{"name":"Departamento de Ingenier\u00eda, Universidad Popular Aut\u00f3noma del Estado de Puebla, Puebla 72410, Mexico"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-1748-9901","authenticated-orcid":false,"given":"J. A.","family":"M\u00e9ndez-Berm\u00fadez","sequence":"additional","affiliation":[{"name":"Departamento de Matem\u00e1tica Aplicada e Estat\u00edstica, Instituto de Ci\u00eancias Matem\u00e1ticas e de Computa\u00e7\u00e3o, Universidade de S\u00e3o Paulo\u2014Campus de S\u00e3o Carlos, Caixa Postal 668, S\u00e3o Carlos 13560-970, Brazil"},{"name":"Instituto de F\u00edsica, Benem\u00e9rita Universidad Aut\u00f3noma de Puebla, Apartado Postal J-48, Puebla 72570, Mexico"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"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":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2020,8,11]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1021\/ja01193a005","article-title":"Structural determination of paraffin boiling points","volume":"69","author":"Wiener","year":"1947","journal-title":"J. Am. Chem. Soc."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"2331","DOI":"10.1246\/bcsj.44.2332","article-title":"Topological index. A newly proposed quantity characterizing the topological nature of structural isomers of saturated hydrocarbons","volume":"44","author":"Hosoya","year":"1971","journal-title":"Bull. Chem. Soc. Jpn."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"535","DOI":"10.1016\/0009-2614(72)85099-1","article-title":"Graph theory and molecular orbitals. 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