{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,8]],"date-time":"2026-01-08T21:09:56Z","timestamp":1767906596083,"version":"3.49.0"},"reference-count":33,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,30]],"date-time":"2020-12-30T00:00:00Z","timestamp":1609286400000},"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>A topic of current interest in the study of topological indices is to find relations between some index and one or several relevant parameters and\/or other indices. In this paper we study two general topological indices A\u03b1 and B\u03b1, defined for each graph H=(V(H),E(H)) by A\u03b1(H)=\u2211ij\u2208E(H)f(di,dj)\u03b1 and B\u03b1(H)=\u2211i\u2208V(H)h(di)\u03b1, where di denotes the degree of the vertex i and \u03b1 is any real number. Many important topological indices can be obtained from A\u03b1 and B\u03b1 by choosing appropriate symmetric functions and values of \u03b1. This new framework provides new tools that allow to obtain in a unified way inequalities involving many different topological indices. In particular, we obtain new optimal bounds on the variable Zagreb indices, the variable sum-connectivity index, the variable geometric-arithmetic index and the variable inverse sum indeg index. Thus, our approach provides both new tools for the study of topological indices and new bounds for a large class of topological indices. We obtain several optimal bounds of A\u03b1 (respectively, B\u03b1) involving A\u03b2 (respectively, B\u03b2). Moreover, we provide several bounds of the variable geometric-arithmetic index in terms of the variable inverse sum indeg index, and two bounds of the variable inverse sum indeg index in terms of the variable second Zagreb and the variable sum-connectivity indices.<\/jats:p>","DOI":"10.3390\/sym13010043","type":"journal-article","created":{"date-parts":[[2020,12,30]],"date-time":"2020-12-30T20:13:41Z","timestamp":1609359221000},"page":"43","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":22,"title":["Mathematical Properties of Variable Topological Indices"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-4352-5109","authenticated-orcid":false,"given":"Jos\u00e9 M.","family":"Sigarreta","sequence":"first","affiliation":[{"name":"Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, Acapulco 39087, Guerrero, Mexico"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,30]]},"reference":[{"key":"ref_1","unstructured":"Gutman, I., and Furtula, B. (2008). Recent Results in the Theory of Randi\u0107 Index, Univ. Kragujevac."},{"key":"ref_2","unstructured":"Li, X., and Gutman, I. (2006). Mathematical Aspects of Randi\u0107 Type Molecular Structure Descriptors, Univ. Kragujevac."},{"key":"ref_3","first-page":"127","article-title":"A survey on the Randi\u0107 index","volume":"59","author":"Li","year":"2008","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1016\/j.akcej.2017.09.006","article-title":"Randi\u0107 index and information","volume":"15","author":"Gutman","year":"2018","journal-title":"AKCE Int. J. 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