Estrada and L-Estrada Indices of Edge-Independent Random Graphs

On Sombor Index

The concept of Sombor index (SO) was recently introduced by Gutman in the chemical graph theory. It is a vertex-degree-based topological index and is denoted by Sombor index SO: SO=SO(G)=∑vivj∈E(G)dG(vi)2+dG(vj)2, where dG(vi) is the degree of vertex vi in G. Here, we present novel lower and upper bounds on the Sombor index of graphs by using some graph parameters. Moreover, we obtain several relations on Sombor index with the first and second Zagreb indices of graphs. Finally, we give some conclusions and propose future work.

Computing Degree Based Topological Properties of Third Type of Hex-Derived Networks

In chemical graph theory, a topological index is a numerical representation of a chemical network, while a topological descriptor correlates certain physicochemical characteristics of underlying chemical compounds besides its chemical representation. The graph plays a vital role in modeling and designing any chemical network. Simonraj et al. derived a new type of graphs, which is named a third type of hex-derived networks. In our work, we discuss the third type of hex-derived networks H D N 3 ( r ) , T H D N 3 ( r ) , R H D N 3 ( r ) , C H D N 3 ( r ) , and compute exact results for topological indices which are based on degrees of end vertices.

Statistically and Relatively Modular Deferred-Weighted Summability and Korovkin-Type Approximation Theorems

The concept of statistically deferred-weighted summability was recently studied by Srivastava et al. . The present work is concerned with the deferred-weighted summability mean in various aspects defined over a modular space associated with a generalized double sequence of functions. In fact, herein we introduce the idea of relatively modular deferred-weighted statistical convergence and statistically as well as relatively modular deferred-weighted summability for a double sequence of functions. With these concepts and notions in view, we establish a theorem presenting a connection between them. Moreover, based upon our methods, we prove an approximation theorem of the Korovkin type for a double sequence of functions on a modular space and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results. Finally, an illustrative example is provided here by the generalized bivariate Bernstein–Kantorovich operators of double sequences of functions in order to demonstrate that our established theorem is stronger than its traditional and statistical versions.

Role of Graphic Integer Sequence in the Determination of Graph Integrity

Networks have an important role in our daily lives. The effectiveness of the network decreases with the breaking down of some vertices or links. Therefore, a less vulnerable communication network is required for greater stability. Vulnerability is the measure of resistance of the network after failure of communication links. In this article, a graph has been taken for modeling a network and integrity as a measure of vulnerability. The approach is to estimate the integrity or upper bound of integrity of at least one connected graph or network constructed from the given graphic integer sequence. Experiments have been done with random graphs, complex networks and also a comparison between two parameters, namely the vertex connectivity and graph integrity as a measure of the network vulnerability have been carried out by removing vertices randomly from various complex networks. A comparison with the existing method shows that the algorithm proposed in this article provides a much better integrity measurement.