On Irregularity Measures of Some Dendrimers Structures

3D molecular structural modeling and characterization of indium phosphide via irregularity topological indices

Indium phosphide (InP) is a binary semiconductor composed of indium and phosphorus. It has a zinc blende crystal structure, which is a type of cubic lattice structure. This lattice is composed of indium and phosphorus atoms arranged in a lattice of cube-shaped cells, with each cell containing four indium atoms and four phosphorus atoms. This lattice structure is the same for all materials with a zinc blende crystal structure and is the most common type of lattice structure in semiconductors. Indium phosphide (InP) has several chemical applications. It is commonly used as a dopant in the production of semiconductors, where it helps control the electrical properties of the material. InP is also utilized in the synthesis various indium-containing compounds, which can have applications in catalysts and chemical reactions. Additionally, InP nanoparticles have been investigated for their potential use in biomedical imaging and drug delivery systems. The topological characterization of 3D molecular structures can be performed via graph theory. In graph theory, the connections between atoms are represented as edges and the atoms themselves are represented as nodes. Furthermore, graph theory can be used to calculate the topological descriptors of the molecule such as the degree-based and reverse degree-based irregularity toplogical indices. These descriptors can be used to compare the topology of different molecules. This paper deals with the modeling and topological characterization of indium phosphide ( InP ) via degree-based and reverse irregularity indices. The 3D crystal structure of the InP is topologically modeled via partition of the edges, and derived closed form expressions for its irregularity indices. Our obtained results will be surely be helpful in investigating the QSPR/QSAR analysis as well as understanding the deep irregular behavior of the indium phosphide ( InP ) .

Distributed Mechanism for Detecting Average Consensus with Maximum-Degree Weights in Bipartite Regular Graphs

In recent decades, distributed consensus-based algorithms for data aggregation have been gaining in importance in wireless sensor networks since their implementation as a complementary mechanism can ensure sensor-measured values with high reliability and optimized energy consumption in spite of imprecise sensor readings. In the presented article, we address the average consensus algorithm over bipartite regular graphs, where the application of the maximum-degree weights causes the divergence of the algorithm. We provide a spectral analysis of the algorithm, propose a distributed mechanism to detect whether a graph is bipartite regular, and identify how to reconfigure the algorithm so that the convergence of the average consensus algorithm is guaranteed over bipartite regular graphs. More specifically, we identify in the article that only the largest and the smallest eigenvalues of the weight matrix are located on the unit circle; the sum of all the inner states is preserved at each iteration despite the algorithm divergence; and the inner states oscillate between two values close to the arithmetic means determined by the initial inner states from each disjoint subset. The proposed mechanism utilizes the first-order forward and backward finite-difference of the inner states (more specifically, five conditions are proposed) to detect whether a graph is bipartite regular or not. Subsequently, the mixing parameter of the algorithm can be reconfigured the way it is identified in this study whereby the convergence of the algorithm is ensured in bipartite regular graphs. In the experimental part, we tested our mechanism over randomly generated bipartite regular graphs, random graphs, and random geometric graphs with various parameters, thereby identifying its very high detection rate and proving that the algorithm can estimate the arithmetic mean with high precision (like in error-free scenarios) after the suggested reconfiguration.

On Ve-Degree-Based Irregularity Properties of the Crystallographic Structure of Molecules

Irregularity indices are usually used for quantitative characterization of the topological structure of nonregular graphs. In numerous applications and problems in material engineering and chemistry, it is useful to be aware that how irregular a molecular structure is? In this paper, we are interested in formulating closed forms of irregularity measures of some of the crystallographic structures of ${\text{Cu}}_2\text{O}\left[p,q,r\right]$ and crystallographic structure of titanium difluoride of $T_i{F}_2\left[p,q,r\right]$ . These theoretical conclusions provide practical guiding significance for pharmaceutical engineering and complex network and quantify the degree of folding of long organic molecules.

Irregularity Measures of Subdivision Vertex-Edge Join of Graphs

The study of graphs and networks accomplished by topological measures plays an applicable task to obtain their hidden topologies. This procedure has been greatly used in cheminformatics, bioinformatics, and biomedicine, where estimations based on graph invariants have been made available for effectively communicating with the different challenging tasks. Irregularity measures are mostly used for the characterization of the nonregular graphs. In several applications and problems in various areas of research like material engineering and chemistry, it is helpful to be well-informed about the irregularity of the underline structure. Furthermore, the irregularity indices of graphs are not only suitable for quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) studies but also for a number of chemical and physical properties, including toxicity, enthalpy of vaporization, resistance, boiling and melting points, and entropy. In this article, we compute the irregularity measures including the variance of vertex degrees, the total irregularity index, the $\sigma$ irregularity index, and the Gini index of a new graph operation.

On Topological Descriptors of Certain Metal-Organic Frameworks

Topological indices are numerical numbers that represent the topology of a molecule and are calculated from the graphical depiction of the molecule. The importance of topological indices is due to their use as descriptors in QSPR/QSAR modeling. QSPRs (quantitative structure-property relationships) and QSARs (quantitative structure-activity relationships) are mathematical correlations between a specified molecular property or biological activity and one or more physicochemical and/or molecular structural properties. In this paper, we give explicit expressions of some degree-based topological indices of two classes of metal-organic frameworks (MOFs), namely, butylated hydroxytoluene- (BHT-) based metal-organic ( $\text{M}=\text{Co}$ , Fe, Mn, Cr) (MBHT) frameworks and ${\text{M}}_1\text{TPyP}-{\text{M}}_2$ (TPyP =  $\mathrm{5,10,15,20}$ -tetrakis(4-pyridyl)porphyrin and ${\text{M}}_1,{\text{M}}_2$  = Fe and Co) MOFs.

On Valency-Based Molecular Topological Descriptors of Subdivision Vertex-Edge Join of Three Graphs

In the studies of quantitative structure–activity relationships (QSARs) and quantitative structure–property relationships (QSPRs), graph invariants are used to estimate the biological activities and properties of chemical compounds. In these studies, degree-based topological indices have a significant place among the other descriptors because of the ease of generation and the speed with which these computations can be accomplished. In this paper, we give the results related to the first, second, and third Zagreb indices, forgotten index, hyper Zagreb index, reduced first and second Zagreb indices, multiplicative Zagreb indices, redefined version of Zagreb indices, first reformulated Zagreb index, harmonic index, atom-bond connectivity index, geometric-arithmetic index, and reduced reciprocal Randić index of a new graph operation named as “subdivision vertex-edge join” of three graphs.

Molecular Irregularity Indices of Nanostar, Fullerene, and Polymer Dendrimers

Dendrimers are highly branched organic macromolecules with successive layers of branch units surrounding a central core. Some properties like toxicity, entropy, and heats of vaporization of these dendrimers can be forecasted using topological indices. The present article is devoted to study of irregularity indices of three well-known classes of dendrimers, namely, nanostar dendrimer D[p], fullerene dendrimer NS4[p], and polymer dendrimerNS5[p], where p is the step size. We also see the relation of irregularity of these dendrimers on the step size graphically.

Irregularity of Block Shift Networks and Hierarchical Hypercube Networks

There is extremely a great deal of mathematics associated with electrical and electronic engineering. It relies upon what zone of electrical and electronic engineering; for instance, there is much increasingly theoretical mathematics in communication theory, signal processing and networking, and so forth. Systems include hubs speaking with one another. A great deal of PCs connected together structure a system. Mobile phone clients structure a network. Networking includes the investigation of the most ideal method for executing a system. Graph theory has discovered a significant use in this zone of research. In this paper, we stretch out this examination to interconnection systems. Hierarchical interconnection systems (HINs) give a system to planning systems with diminished connection cost by exploiting the area of correspondence that exists in parallel applications. HINs utilize numerous levels. Lower-level systems give nearby correspondence, while more significant level systems encourage remote correspondence. HINs provide issue resilience within the sight of some defective nodes and additionally interfaces. Existing HINs can be comprehensively characterized into two classes: those that use nodes or potential interface replication and those that utilize reserve interface nodes.

Computational Analysis of Imbalance-Based Irregularity Indices of Boron Nanotubes

. Molecular topology provides a basis for the correlation of physical as well as chemical properties of a certain molecule. Irregularity indices are used as functions in the statistical analysis of the topological properties of certain molecular graphs and complex networks, and hence help us to correlate properties like enthalpy, heats of vaporization, and boiling points etc. with the molecular structure. In this article we are interested in formulating closed forms of imbalance-based irregularity measures of boron nanotubes. These tubes are known as α-boron nanotube, triangular boron nanotubes, and tri-hexagonal boron nanotubes. We also compare our results graphically and come up with the conclusion that alpha boron tubes are the most irregular with respect to most of the irregularity indices.

Comparison of Irregularity Indices of Several Dendrimers Structures

Irregularity indices are usually used for quantitative characterization of the topological structures of non-regular graphs. In numerous problems and applications, especially in the fields of chemistry and material engineering, it is useful to be aware of the irregularity of a molecular structure. Furthermore, the evaluation of the irregularity of graphs is valuable not only for quantitative structure-property relationship (QSPR) and quantitative structure-activity relationship (QSAR) studies but also for various physical and chemical properties, including entropy, enthalpy of vaporization, melting and boiling points, resistance, and toxicity. In this paper, we will restrict our attention to the computation and comparison of the irregularity measures of different classes of dendrimers. The four irregularity indices which we are going to investigate are σ irregularity index, the irregularity index by Albertson, the variance of vertex degrees, and the total irregularity index.

Imbalance-Based Irregularity Molecular Descriptors of Nanostar Dendrimers

Dendrimers are branched organic macromolecules with successive layers of branch units surrounding a central core. The molecular topology and the irregularity of their structure plays a central role in determining structural properties like enthalpy and entropy. Irregularity indices which are based on the imbalance of edges are determined for the molecular graphs associated with some general classes of dendrimers. We also provide graphical analysis of these indices for the above said classes of dendrimers.

Irregularity Molecular Descriptors of Hourglass, Jagged-Rectangle, and Triangular Benzenoid Systems

Determining the degree of irregularity of a certain molecular structure or a network has been a key source of interest for molecular topologists, but it is also important as it provides an insight into the key features used to guess properties of the structures. In this article, we are interested in formulating closed forms of irregularity measures of some popular benzenoid systems, such as hourglass H (m, n), jagged-rectangular J (m, n), and triangular benzenoid T (m, n) systems. We also compared our results graphically and concluded which benzenoid system among the above listed is more irregular than the others.