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Computational and analytical studies of the harmonic index on Erdös-Rényi models

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dc.affiliation.dptoUC3M. Departamento de Matemáticases
dc.affiliation.grupoinvUC3M. Grupo de Investigación: Análisis Aplicadoes
dc.contributor.authorMartinez Martinez, C.T.
dc.contributor.authorMendez Bermudez, J.A.
dc.contributor.authorRodríguez García, José Manuel
dc.contributor.authorSigarreta Almira, José María
dc.contributor.funderMinisterio de Ciencia, Innovación y Universidades (España)es
dc.date.accessioned2023-01-17T11:38:09Z
dc.date.available2023-01-17T11:38:09Z
dc.date.issued2021-04
dc.description.abstractA main topic in the study of topological indices is to find bounds of the indices involving several parameters and/or other indices. In this paper we perform statistical (numerical) and analytical studies of the harmonic index H(G), and other topological indices of interest, on Erdos-Rényi (ER) graphs G(n, p) characterized by n vertices connected independently with probability p ∈ (0, 1). Particularly, in addition to H(G), we study here the (−2) sum-connectivity index χ−2(G), the modified Zagreb index MZ(G), the inverse degree index ID(G) and the Randic index R(G). First, to perform the statistical study of these indices, we define the averages of the normalized indices to their maximum value: {H(G)}, {χ−2(G)}, {MZ(G)}, {ID(G)}, {R(G)}. Then, from a detailed scaling analysis, we show that the averages of the normalized indices scale with the product ξ ≈ np. Moreover, we find two different behaviors. On the one hand, hH(G)i and hR(G)i, as a function of the probability p, show a smooth transition from zero to n/2 as p increases from zero to one. Indeed, after scaling, it is possible to define three regimes: a regime of mostly isolated vertices when ξ < 0.01 (H(G), R(G) ≈ 0), a transition regime for 0.01 < ξ < 10 (where 0 < H(G), R(G) < n/2), and a regime of almost complete graphs for ξ > 10 (H(G), R(G) ≈ n/2). On the other hand, hχ−2(G)i, hMZ(G)i and hID(G)i increase with p until approaching their maximum value, then they decrease by further increasing p. Thus, after scaling the curves corresponding to these indices display bell-like shapes in log scale, which are symmetric around ξ ≈ 1; i.e. the percolation transition point of ER graphs. Therefore, motivated by the scaling analysis, we analytically (i) obtain new relations connecting the topological indices H, χ−2, MZ, ID and R that characterize graphs which are extremal with respect to the obtained relations and (ii) apply these results in order to obtain inequalities on H, χ−2, MZ, ID and R for graphs in ER models.en
dc.description.sponsorshipJ.A.M.-B. acknowledges financial support from FAPESP (Grant No. 2019/ 06931-2), Brazil, CONACyT (Grant No. 2019-000009-01EXTV-00067) and PRODEP-SEP (Grant No. 511-6/2019.-11821), Mexico. J.M.R. and J.M.S. acknowledge financial support from Agencia Estatal de Investigación (PID2019-106433GB-I00/AEI/ 10.13039/501100011033), Spain.en
dc.format.extent32
dc.identifier.bibliographicCitationMartínez-Martínez, C. T., Méndez-Bermúdez, A., Rodrígez, José M. & Sigarreta, José M.(2021). Computational and analytical studies of the harmonic index on Erdös-Rényi models. MATCH Communications in Mathematical and in Computer Chemistry 85(2), 395-426.en
dc.identifier.issn0340-6253
dc.identifier.otherhttps://match.pmf.kg.ac.rs/content85n2.htm
dc.identifier.publicationfirstpage395
dc.identifier.publicationissue2
dc.identifier.publicationlastpage426
dc.identifier.publicationtitleMATCH Communications in Mathematical and in Computer Chemistryen
dc.identifier.publicationvolume85
dc.identifier.urihttps://hdl.handle.net/10016/36283
dc.identifier.uxxiAR/0000030820
dc.language.isoengen
dc.publisherMATCH
dc.relation.projectIDGobierno de España. PID2019-106433GB-I00es
dc.relation.publisherversionhttps://match.pmf.kg.ac.rs/electronic_versions/Match85/n2/match85n2_395-426.pdf
dc.rights© match, 2021
dc.rights.accessRightsopen accessen
dc.subject.ecienciaMatemáticases
dc.titleComputational and analytical studies of the harmonic index on Erdös-Rényi modelsen
dc.typeresearch article*
dc.type.hasVersionVoR*
dspace.entity.typePublication
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