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Author(s): 
MartinezMartinez, C. T.
^{[1, 2]}
;
MendezBermudez, J. A.
^{[2, 3]}
;
Rodriguez, Jose M.
^{[4]}
;
Sigarreta, Jose M.
^{[5]}
Total Authors: 4

Affiliation:  ^{[1]} Univ Zaragoza, Inst Biocomputat & Phys Complex Syst BIFI, Zaragoza 50018  Spain
^{[2]} Benemerita Univ Autonoma Puebla, Inst Fis, Apartado Postal J48, Puebla 72570  Mexico
^{[3]} Univ Sao Paulo, Inst Ciencias Matemat & Comp, Dept Matemat Aplicada & Estat, Campus Sao Carlos, Caixa Postal 668, BR13560970 Sao Carlos, SP  Brazil
^{[4]} Univ Carlos III Madrid, Dept Matemat, Ave Univ 30, Madrid 28911  Spain
^{[5]} Univ Autonoma Guerrero, Acapulco De Juarez 39610, Guerrero  Mexico
Total Affiliations: 5

Document type:  Journal article 
Source:  MATCHCOMMUNICATIONS IN MATHEMATICAL AND IN COMPUTER CHEMISTRY; v. 85, n. 2, p. 395426, 2021. 
Web of Science Citations:  1 
Abstract  
A 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 ErdosRenyi (ER) graphs G(n, p) characterized by n vertices connected independently with probability p is an element of (0,1). Particularly, in addition to H (G), we study here the (2) sumconnectivity index chi2(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) over bar (G)>, <(chi) over bar (2)(G), <(MZ) over bar (G)>, <(ID) over bar (G)> and <(R) over bar (G)>. Then, from a detailed scaling analysis, we show that the averages of the normalized indices scale with the product np. Moreover, we find two different behaviors. On the one hand, < H(G)> and < R(G)>, 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 xi < 0.01 (H(G), R(G) approximate to 0), a transition regime for 0.01 < xi < 10 (where 0 < H(G), R(G) < n/2), and a regime of almost complete graphs for xi > 10 (H(G), R(G) approximate to n/2). On the other hand, <chi(2)(G)>, < MZ(G)> and < ID(G)> 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 belllike shapes in log scale, which are symmetric around xi = 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, chi(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, chi(2), MZ, ID and R for graphs in ER models. (AU)  
FAPESP's process:  19/069312  Random matrix theory approach to complex networks 
Grantee:  Francisco Aparecido Rodrigues 
Support type:  Research Grants  Visiting Researcher Grant  International 