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Computational and Analytical Studies of the Harmonic Index on Erdos-Renyi Models

Martinez-Martinez, C. T. [1, 2] ; Mendez-Bermudez, J. A. [2, 3] ; Rodriguez, Jose M. [4] ; Sigarreta, Jose M. [5]
Número total de Autores: 4
Afiliação do(s) autor(es):
[1] Univ Zaragoza, Inst Biocomputat & Phys Complex Syst BIFI, Zaragoza 50018 - Spain
[2] Benemerita Univ Autonoma Puebla, Inst Fis, Apartado Postal J-48, Puebla 72570 - Mexico
[3] Univ Sao Paulo, Inst Ciencias Matemat & Comp, Dept Matemat Aplicada & Estat, Campus Sao Carlos, Caixa Postal 668, BR-13560970 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
Número total de Afiliações: 5
Tipo de documento: Artigo Científico
Citações Web of Science: 1

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 Erdos-Renyi (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) sum-connectivity index chi-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) 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 bell-like 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)

Processo FAPESP: 19/06931-2 - Métodos de matrizes aleatórias em redes complexas
Beneficiário:Francisco Aparecido Rodrigues
Linha de fomento: Auxílio à Pesquisa - Pesquisador Visitante - Internacional