Normalized Sombor Indices as Complexity Measures of Random Networks.

We perform a detailed computational study of the recently introduced Sombor indices on random networks. Specifically, we apply Sombor indices on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. Within a statistical random matrix theory approach, we show that the average values of Sombor indices, normalized to the order of the network, scale with the average degree. Moreover, we discuss the application of average Sombor indices as complexity measures of random networks and, as a consequence, we show that selected normalized Sombor indices are highly correlated with the Shannon entropy of the eigenvectors of the adjacency matrix.

Scaling properties of the action in the Riemann-Liouville fractional standard map.

The Riemann-Liouville fractional standard map (RL-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables (I,θ). The RL-fSM is parameterized by K and α∈(1,2], which control the strength of nonlinearity and the fractional order of the Riemann-Liouville derivative, respectively. In this work we present a scaling study of the average squared action 〈I^{2}〉 of the RL-fSM along strongly chaotic orbits, i.e., for K≫1. We observe two scenarios depending on the initial action I_{0}, I_{0}≪K or I_{0}≫K. However, we can show that 〈I^{2}〉/I_{0}^{2} is a universal function of the scaled discrete time nK^{2}/I_{0}^{2} (n being the nth iteration of the RL-fSM). In addition, we note that 〈I^{2}〉 is independent of α for K≫1. Analytical estimations support our numerical results.

On the variable inverse sum deg index.

Several important topological indices studied in mathematical chemistry are expressed in the following way $ \sum_{uv \in E(G)} F(d_u, d_v) $, 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 $ d_u $ is the degree of the vertex $ u $. Among them, the variable inverse sum deg index $ IS\!D_a $, with $ F(d_u, d_v) = 1/(d_u^a+d_v^a) $, was found to have several applications. In this paper, we solve some problems posed by Vukicevic [1], and we characterize graphs with maximum and minimum values of the $ IS\!D_a $ 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 hydrocarbons.

Revan Sombor indices: Analytical and statistical study.

In this paper, we perform analytical and statistical studies of Revan indices on graphs $ G $: $ R(G) = \sum_{uv \in E(G)} F(r_u, r_v) $, where $ uv $ denotes the edge of $ G $ connecting the vertices $ u $ and $ v $, $ r_u $ is the Revan degree of the vertex $ u $, and $ F $ is a function of the Revan vertex degrees. Here, $ r_u = \Delta + \delta - d_u $ with $ \Delta $ and $ \delta $ the maximum and minimum degrees among the vertices of $ G $ and $ d_u $ is the degree of the vertex $ u $. We concentrate on Revan indices of the Sombor family, i.e., the Revan Sombor index and the first and second Revan $ (a, b) $-$ KA $ indices. First, we present new relations to provide bounds on Revan Sombor indices which also relate them with other Revan indices (such as the Revan versions of the first and second Zagreb indices) and with standard degree-based indices (such as the Sombor index, the first and second $ (a, b) $-$ KA $ indices, the first Zagreb index and the Harmonic index). Then, we extend some relations to index average values, so they can be effectively used for the statistical study of ensembles of random graphs.

Extremal problems on exponential vertex-degree-based topological indices.

In this work we obtain new lower and upper optimal bounds for general (exponential) indices of a graph. In the same direction, we show new inequalities involving some well-known topological indices like the generalized atom-bound connectivity index $ ABC_\alpha $ and the generalized second Zagreb index $ M_2^\alpha $. Moreover, we solve some extremal problems for their corresponding exponential indices ($ e^{ABC_\alpha} $ and $ e^{M_2^{\alpha}} $).

Analytical and computational properties of the variable symmetric division deg index.

The aim of this work is to obtain new inequalities for the variable symmetric division deg index $ SDD_\alpha(G) = \sum_{uv \in E(G)} (d_u^\alpha/d_v^\alpha+d_v^\alpha/d_u^\alpha) $, and to characterize graphs extremal with respect to them. Here, by $ uv $ we mean the edge of a graph $ G $ joining the vertices $ u $ and $ v $, and $ d_u $ denotes the degree of $ u $, and $ \alpha \in \mathbb{R} $. Some of these inequalities generalize and improve previous results for the symmetric division deg index. In addition, we computationally apply the $ SDD_\alpha(G) $ index on random graphs and we demonstrate that the ratio $ \langle SDD_\alpha(G) \rangle/n $ ($ n $ is the order of the graph) depends only on the average degree $ \langle d \rangle $.

Topological versus spectral properties of random geometric graphs.

In this work we perform a detailed statistical analysis of topological and spectral properties of random geometric graphs (RGGs), a graph model used to study the structure and dynamics of complex systems embedded in a two-dimensional space. RGGs, G(n,ℓ), consist of n vertices uniformly and independently distributed on the unit square, where two vertices are connected by an edge if their Euclidian distance is less than or equal to the connection radius ℓ∈[0,sqrt[2]]. To evaluate the topological properties of RGGs we chose two well-known topological indices, the Randic index R(G) and the harmonic index H(G). We characterize the spectral and eigenvector properties of the corresponding randomly weighted adjacency matrices by the use of random matrix theory measures: the ratio between consecutive eigenvalue spacings, the inverse participation ratios, and the information or Shannon entropies S(G). First, we review the scaling properties of the averaged measures, topological and spectral, on RGGs. Then we show that (i) the averaged-scaled indices, 〈R(G)〉 and 〈H(G)〉, are highly correlated with the average number of nonisolated vertices 〈V_{×}(G)〉; and (ii) surprisingly, the averaged-scaled Shannon entropy 〈S(G)〉 is also highly correlated with 〈V_{×}(G)〉. Therefore, we suggest that very reliable predictions of eigenvector properties of RGGs could be made by computing topological indices.