Inverse Sum Research Articles

On Analysis of Topological Aspects for Subdivision of Kragujevac Tree Networks

In this paper, we have taken review of certain topological topological characteristics of subdivision and the line graph of subdivision of Kragujevac tree. A Kragujevac tree is denoted by K , K ∈ Kg q = r 2 t + 1 + 1 , r , with order r 2 t + 1 + 1 and size r 2 t + 1 , respectively. We have computed the Zagreb polynomials, forgotten polynomial, and M-polynomial for Kragujevac tree. Moreover, we have computed topological indices like Zagreb-type indices, reduced reciprocal Randić indices, family of Gourava indices as well as forgotten index. Further, some topological indices that can be directly derived from M-polynomial, i.e., first and second Zagreb index, modified second Zagreb index, Randić and reciprocal Randić index, symmetric division and harmonic index, and inverse sum and augmented Zagreb index are also computed.

On Computation Degree-Based Topological Descriptors for Planar Octahedron Networks

A molecular graph is used to represent a chemical molecule in chemical graph theory, which is a branch of graph theory. A graph is considered to be linked if there is at least one link between its vertices. A topological index is a number that describes a graph’s topology. Cheminformatics is a relatively young discipline that brings together the field of sciences. Cheminformatics helps in establishing QSAR and QSPR models to find the characteristics of the chemical compound. We compute the first and second modified K-Banhatti indices, harmonic K-Banhatti index, symmetric division index, augmented Zagreb index, and inverse sum index and also provide the numerical results.

Vertex-Based Topological Indices of Double and Strong Double Graph of Dutch Windmill Graph

A measurement of the molecular topology of graphs is known as a topological index, and several physical and chemical properties such as heat formation, boiling point, vaporization, enthalpy, and entropy are used to characterize them. Graph theory is useful in evaluating the relationship between various topological indices of some graphs derived by applying certain graph operations. Graph operations play an important role in many applications of graph theory because many big graphs can be obtained from small graphs. Here, we discuss two graph operations, i.e., double graph and strong double graph. In this article, we will compute the topological indices such as geometric arithmetic index GA , atom bond connectivity index ABC , forgotten index F , inverse sum indeg index ISI , general inverse sum indeg index ISI α , β , first multiplicative-Zagreb index PM 1 and second multiplicative-Zagreb index PM 2 , fifth geometric arithmetic index GA 5 , fourth atom bond connectivity index ABC 4 of double graph, and strong double graph of Dutch Windmill graph D 3 p .

Computing SS Index of Certain Dendrimers

The numerical descriptor gathers the data from the molecular graphs and helps to know the characteristics of the chemical structure known as topological index. The QSAR/QSPR/QSTR studies are benefited with the significant role played by topological indices in the drug design. Topological indices provide the information about the physical/chemical/biological properties of chemical compounds. The Zagreb indices are widely studied because of their extensive usage in chemical graph theory. Inspired by the earlier work on inverse sum indeg index (ISI index), novel topological index known as SS index is introduced and computed for four dendrimer structures. Also, the strong correlation coefficient between SS index and 5 physico-chemical characteristics such as boiling point (bp), molar volume (mv), molar refraction (mr), heats of vaporization (hv), and critical pressure (cp) of 67 alkane isomers have been determined. It is found that newly introduced index has shown good correlation in comparison with three most popular existing indices (ISI index and first and second Zagreb indices). In the last part, the mathematical properties of SS index are discussed.

Molecular Descriptor Analysis of Certain Isomeric Natural Polymers

Polymers, drugs, and almost all chemical or biochemical compounds are frequently modeled as diverse ω -cyclic, acyclic, bipartite, and polygonal shapes and regular graphs. Molecular descriptors (topological indices) are the numerical quantities and computed from the molecular graph Γ (2D lattice). These descriptors are highly significant in quantitative structure-property or activity relationship (QSPR and QSAR) modeling that provides the theoretical and the optimal basis to expensive experimental drug design. In this paper, we study three isomeric natural polymers of glucose (polysaccharides), namely, cellulose, glycogen, and amylopectin (starch), having promising pharmaceutical applications, exceptional properties, and fascinating molecular structures. We intend to investigate and compute various closed-form formulas such as ABC , GA , sum-connectivity χ − 1 / 2 , ABC 4 , GA 5 , and Sanskruti indices for the aforementioned macromolecules. Also, we present the closed-form formulas for the first, second, modified, and augmented Zagreb indices, inverse and general Randić indices, and symmetric division deg, harmonic, and inverse sum indices. Furthermore, we provide a comparative analysis using 3D graphs for these families of macromolecules to clarify their nature.

A note on the minimum inverse sum indeg index of cacti

Let G=(V,E) be a simple graph. The inverse sum indeg index of G is defined as ISI(G)=∑uv∈E(G)d(u)d(v)d(u)+d(v), where d(u) is the degree of u. This index has a nice predicting ability for the total surface area of octane isomers. A cactus is a connected graph in which all of its blocks are either edges or cycles. Let Cn,k be the set of cacti of order n and with k cycles. In this note, we characterize all cacti with minimum inverse sum indeg index among Cn,k when n≥6k−3.

Inverse Sum Status Energy of a Graph

Motivated by the inverse sum status index, we introduce the inverse sum status matrix ISS={■((σ_u σ_v)/((σ_u+σ_v ) ) if u_i~v_j,@0 otherwise)┤ Thus we also obtained the results for well known graphs. Keywords: Inverse sum status energy, Inverse Sum Indeg matrix. 2010 AMS Subject Classification: 05C50.

Sharp Bounds for the Inverse Sum Indeg Index of Graph Operations

Vukičević and Gasperov introduced the concept of 148 discrete Adriatic indices in 2010. These indices showed good predictive properties against the testing sets of the International Academy of Mathematical Chemistry. Among these indices, twenty indices were taken as beneficial predictors of physicochemical properties. The inverse sum indeg index denoted by ISI G k of G k is a notable predictor of total surface area for octane isomers and is presented as ISI G k = ∑ g k g k ′ ∈ E G k d G k g k d G k g k ′ / d G k g k + d G k g k ′ , where d G k g k represents the degree of g k ∈ V G k . In this paper, we determine sharp bounds for ISI index of graph operations, including the Cartesian product, tensor product, strong product, composition, disjunction, symmetric difference, corona product, Indu–Bala product, union of graphs, double graph, and strong double graph.

Bounds for the Topological Indices of ℘ graph

Topological indices are mathematical measure which correlates to the chemical structures of any simple finite graph. These are used for Quantitative Structure-Activity Relationship (QSAR) and Quantitative Structure-Property Relationship (QSPR). In this paper, we define operator graph namely, ℘ graph and structured properties. Also, establish the lower and upper bounds for few topological indices namely, Inverse sum indeg index, Geometric-Arithmetic index, Atom-bond connectivity index, first zagreb index and first reformulated Zagreb index of ℘-graph.

Application of Topological Indices of Tenofovir Chemical Structures for the Cure of HIV/AIDS Patients

Human immune deficiency virus (HIV), a retrovirus, is the main reason for acquired immune deficiency syndrome (AIDS), and one of the prime social and medical problems at the present time. Approximately around 100 million citizens all over the world are suffering from HIV virus and around 50 million citizens are dead. This virus expanded rapidly all over the world. Unfortunately, there is no medicine, drug or vaccine is identified to treat this virus. Scientists have recognised the effectiveness of already existing anti-viral drugs to cure and control the HIV virus. Some of them are Tri-POC Tenofovir Dimer, Tenofovir Disproxil and Tenofovir Alafenamide.Topological indices- mathematical elucidations for a molecule can be created by an algorithm and can be applied to its representation. To generate various biological and physicochemical properties of chemical compounds, topological indices are used. In the current study, by using polynomial approach, for the above mentioned antiviral drugs, certain neighbourhood sum& degree based topological indices have been studied. The outcomes achieved can support the invention of new medicine for the cure of AIDS. In the present study, we establish some topological properties of Tenofovir dimer, Tenofovir disproxil and alafenamide used to inhibit the outbreak of AIDS. We compute some topological indices (general indices (Randic and harmonic), Zagreb index (1st, 2nd, 3rd, second modified, Redefined 3rdand Augmented), Forgotten index, Systematic division index and Inverse sum indices for these three chemical structures. In Medical Science, topological index calculation defines the topological index related to molecular structure and its corresponding biological, medical, pharmaceutical, and chemical properties of the medicines. From last twenty to thirty years the world is fronting the danger of identifying cure for AIDS. Nearly 10 million people are getting affected with this disease every year. In order to cure this malicious disease antiviral drugs, in form of anti-metabolites, hormones and alkylating agents are used. As per numerous examinations conducted with their chemical structures, it revealed that there is a association between the characteristic features of these anti-viral medicines, drugs and alkane’s viz.enthalpy, melting point and boiling point. In this current study, numerous topological indices have been defined on few of the above mentioned drugs so as to assist the scientists to identify the chemical reactions and physical & chemical characteristics and biological activities that are related with them. Hence, the topological indices study on the molecular structure of the medicines, drugs chemical compounds, can constitute for absence of laboratory research which provides a notional origin for the production of chemical materials and drugs. In the present study, we analysed the Tenofovir family chemicals which is extensively used in antiviral medicines and drugs invention. Eleven topological indices are analysed using multi-order polynomials and these results can be used in medicine and pharmaceutical experiments thus paving the new way for new drug invention for AIDS.

An identity for the sum of inverses of odd divisors of n in terms of the number of representations of n as a sum of squares

Let $$\sum_{\substack{d|n\\ d\equiv 1 (2)}}\frac{1}{d}$$ denote the sum of inverses of odd divisors of a positive integer $n$, and let $c_{r}(n)$ be the number of representations of $n$ as a sum of $r$ squares where representations with different orders and different signs are counted as distinct. The aim is of this note is to prove the following interesting combinatorial identity: $$ \sum_{\substack{d|n\\ d\equiv 1 (2)}}\frac{1}{d}=\frac{1}{2}\,\sum_{r=1}^{n}\frac{(-1)^{n+r}}{r}\,\binom{n}{r}\, c_{r}(n). $$

Neighbourhood Degree – Based Topological Indices of Graphene Structure

The theory of chemical reaction networks is a branch of mathematics that aims to mimic real-world behavior. This research area has drawn many researchers' attention, primarily due to its biological and empirical chemistry applications. The fascinating problems that emerge from the mathematical structures involved have kindled the interest of pure mathematicians. In this paper, we estimate a few topological indices such as SK index, SK1 index, SK2 index, Modified Randić index, and Inverse Sum Index for the Graphene structure based on the neighborhood degree and obtain results based on both sum and products of the cardinality of edge partitions corresponding to 4 different Graphene structures. We also present the 3D representations of the indices using MATLAB.

Comparative Study of Certain Synthetic Polymers via Bond-Additive Invariants

Polymers, like vulcanized rubber, bakelite, and poly-methyl methacrylate (PMMA), are widely utilized as denture based materials, and their prominence has been nothing short of excellent. Recently, Ahmad et al. [Open Chemistry 17(2019): 663-670] computed bond-additive invariants (BAIs) for the molecular graph of bakelite. In the same paper, they proposed the comparative study of the aforementioned-polymers using BAIs. This paper develops molecular graphs of vulcanized rubber and PMMA to estimate M-Polynomial and the generalized Zagreb index. We derive numerous BAIs such as the first and the second Zagreb, Re-defined Zagreb, general Randić, first general Zagreb, and symmetric division degree invariants from the generalized Zagreb index. Moreover, we obtain the modified second Zagreb, inverse Randić, harmonic, inverse sum, and augmented Zagreb invariants from the M-Polynomials. Besides, we compute the atom bond connectivity, its fourth version ABC <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> , the geometric arithmetic, its fifth version GA <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">5</sub> , and the Sanskruti indices. Finally, we provide insight into the numerical comparison among several BAIs to establish a relation for the underlying polymers' various physicochemical properties.

The Analysis of Spatial-Temporal Evolution of City Accessibility Based on Highway Network in Henan Province in China

Accessibility is an important tool to evaluate the maturity of a regional traffic network structure which describes the traffic convenience in the traffic network. The paper defines a new accessibility index by using the resident population weighted average value of the sum of inverse of the traveling time distance and time threshold coming from ordinary traffic network, and then uses this accessibility index to analyze the spatial-temporal characteristics of Henan highway network, as well as its evolution patterns from 2005 to 2020. The results show that with the expansion and improvement of Henan highway network, city accessibility level has been significantly improved, spatial convergence is obvious, the cities in the north central are always High-High aggregation area, the cities in the south are always Low-Low aggregation area, gradually forming the characteristics of Northwest high and Southeast low, relative balance between East and West. There is some non-conforming phenomenon in highway mileage growth and improvement of the city accessibility levels, but this situation is being weakened, the highway network layout is gradually rationalized, the spatial distribution of city accessibility and that of population are beginning to converge.

Degree based descriptors of certain classes of dendrimer graphs

The degree based topological descriptors are correlated with physico-chemical properties such as stability, strain energy, boiling point etc. of certain chemical compounds. Among the degree based topological descriptors Sanskruti index and inverse sum indeg index are significant predictor of total surface area of some chemical compounds. In this paper, we present the formulae for Sanskruti index and second version of inverse sum indeg index of some classes of dendrimers molecular graphs.

Two graph invariants of certain types of convex polytopes

Abstract A topological invariant is a numerical value which can be associated with a graph structure. These invariant graphs are utilized for modelling information of molecules in structural chemistry and biology. Over the years, many topological indices proposed and studied based on degree, distance and other parameters of the graph. In this paper, we present the several formulae for Sanskruti index. Also, we study the second version of inverse sum indeg index for some types of convex polytopes.

Some inequalities on the inverse sum indeg coindex

Some inequalities on the inverse sum indeg coindex

Some degree-based topological indices of caboxy-terminated dendritic macromolecule

Abstract In the modern era of the chemical science, the chemical graph theory has contributed significantly to exploring the properties of the chemical compounds. Currently, the computation of the topological indices is one of the most active directions of the research in the area of the chemical graph theory. The main feature of the study of the topological indices is its its ability of predicting the various physio-chemical properties. In this article, we compute several degree-based topological indices for the caboxy-terminated dendritic macromolecule. We compute Harmonic index, atom-bond connectivity index, geometric arithmetic index, sum connectivity index, inverse sum index, symmetric division degree, and Zagreb indices for caboxy-terminated dendritic macromolecule. The obtained results have potential to predict biochemical properties such as viscosity, entropy, and boiling point.

The expected values of some indices in random phenylene chains

A special class of conjugated hydrocarbons is known as phenylenes, which is composed of a special arrangement of six- and four-membered rings. In particular, any two six-membered rings (hexagons) are not adjacent, and every four-membered ring (square) is adjacent to a pair of nonadjacent hexagons. If each hexagon of phenylene is adjacent only to two squares, then the obtained chain is called the phenylene chain. The main object of this paper is to determine the expected values of the sum-connectivity, harmonic, symmetric division, variable inverse sum degree and general Randic indices for this class of conjugated hydrocarbons. The comparisons between the expected values of these indices with respect to the random phenylene chains have been determined explicitly. The graphical illustrations have been given in terms of the differences between the expected values of these indices.

Extension of bridge graphs and their chemical applications

Abstract A topological invariant is a numerical value which can be associated with a graph structure. These graph invariants are utilized for modeling information of molecules in structural chemistry and biology. Over the years many topological indices are proposed and studied based on degree, distance and other parameters of graph. Also graph invariants play a significant role in chemistry, especially, QSPR/QSAR studies. In this paper, we concentrate two graph invariants such as augmented Zagreb invariant and inverse sum indeg invariant of extended bridge graphs.