On the variable inverse sum deg index

Several important topological indices studied in mathematical chemistry are expressed in the following way Puv∈E(G) F(du, dv), where F is a two variable function that satisfies the condition F(x, y) = F(y, x), uv denotes an edge of the graph G and du is the degree of the vertex u. Among them, the variable inverse sum deg index ISDa, with F(du, dv) = 1/(dua + dva), was found to have several applications. In this paper, we solve some problems posed by Vukičević [1], and we characterize graphs with maximum and minimum values of the ISDa index, for a < 0, in the following sets of graphs with n vertices: graphs with fixed minimum degree, connected graphs with fixed minimum degree, graphs with fixed maximum degree, and connected graphs with fixed maximum degree. Also, we performed a QSPR analysis to test the predictive power of this index for some physicochemical properties of polyaromatic hydrocarbon

Topological indices for the antiregular graphs

We determine some classical distance-based and degree-based topo- logical indices of the connected antiregular graphs (maximally irregular graphs). More precisely, we obtain explicitly the k-Wiener index, the hyper-Wiener index, the degree distance, the Gutman index, the first, sec- ond and third Zagreb index, the reduced first and second Zagreb index, the forgotten Zagreb index, the hyper-Zagreb index, the refined Zagreb index, the Bell index, the min-deg index, the max-deg index, the symmet- ric division index, the harmonic index, the inverse sum indeg index, the M-polynomial and the Zagreb polynomial

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

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-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 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.J.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

ISI spectral radii and ISI energies of graph operations

Graph energy is defined to be the p-norm of adjacency matrix associated to the graph for p = 1 elaborated as the sum of the absolute eigenvalues of adjacency matrix. The graph’s spectral radius represents the adjacency matrix’s largest absolute eigenvalue. Applications for graph energies and spectral radii can be found in both molecular computing and computer science. On similar lines, Inverse Sum Indeg, (ISI) energies, and (ISI) spectral radii can be constructed. This article’s main focus is the ISI energies, and ISI spectral radii of the generalized splitting and shadow graphs constructed on any regular graph. These graphs can be representation of many physical models like networks, molecules and macromolecules, chains or channels. We actually compute the relations about the ISI energies and ISI spectral radii of the newly created graphs to those of the original graph

Asymptotic distribution of degree--based topological indices

Topological indices play a significant role in mathematical chemistry. Given a graph $\mathcal{G}$ with vertex set $\mathcal{V}=\{1,2,\dots,n\}$ and edge set $\mathcal{E}$, let $d_i$ be the degree of node $i$. The degree-based topological index is defined as $\mathcal{I}_n=$ $\sum_{\{i,j\}\in \mathcal{E}}f(d_i,d_j)$, where $f(x,y)$ is a symmetric function. In this paper, we investigate the asymptotic distribution of the degree-based topological indices of a heterogeneous Erd\H{o}s-R\'{e}nyi random graph. We show that after suitably centered and scaled, the topological indices converges in distribution to the standard normal distribution. Interestingly, we find that the general Randi\'{c} index with $f(x,y)=(xy)^{\tau}$ for a constant $\tau$ exhibits a phase change at $\tau=-\frac{1}{2}$

Билтен Фонда САНУ за истраживања у науци и уметности за 2019. годину . Бр. 45

Фонд САНУ за истраживања у науци и уметности је у 2019. го- дини остварио укупни приход у износу од 60.600.000,00 динара, од Министарства просвете, науке и технолошког развоја Републике Србије (Уг. бр. 1/6 од 22. 3. 2019). Одлуку о расподели средстава за 2019. годину донео је Управни одбор Фонда, на седници одржаној 17. јуна 2019. годи- не, на основу предлога и сугестија Академијиних одељења и Извршног одбора САНУ. Фонд САНУ је уплаћена средства распоредио на следеће позиције: за научноистраживачке и пројекте у области уметности, објављивање на- учних публикација, међународну научну сарадњу, организовање научних скупова, учешће чланова САНУ и њихових сарадника на научним скупо- вима у земљи и иностранству, набавку научне литературе и за друге на- мене Програма научноистраживачких и уметничких активности САНУ. На ове намене утрошен је укупан износ уплаћених средстава, односно 60.600.000,00 динара.До броја 40 часопис је излазио под називом: Билтен Фонда за научна истраживања САНУ (ISSN 0352-2385)