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

publication date

  • April 2021

start page

  • 395

end page

  • 426


  • 2


  • 85

International Standard Serial Number (ISSN)

  • 0340-6253


  • 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 Erd ̋os-R ́enyi (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 M Z(G), the inverse degree index ID(G) and the Randi ́c
    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)〉,〈M Z(G)〉, 〈ID(G)〉 and 〈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, 〈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 ξ 10 (H(G), R(G) ≈ n/2). On the other hand, 〈χ−2(G)〉, 〈M Z(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 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, M Z, 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, M Z, ID and R for graphs in ER models.


  • Mathematics