Mathematical Chemistry Research Articles

On the Merrifield–Simmons index of tricyclic graphs

The total number of independent subsets, including the empty set, of a graph, is also termed as the Merrifield–Simmons index (MSI) in mathematical chemistry. Zhu and Yu (2012) presented a lower bound on Merrifield–Simmons index of tricyclic graphs in terms of order n and the characterization of extremal graphs. This result was erroneous. In this paper, we correct this result. Moreover, we characterize the tricyclic graphs on n vertices with the second and third smallest values of the Merrifield–Simmons index.

Mathematical Chemistry Approaches for Computer-Aid Design of Free-Shaped Structures in Architecture and Construction Engineering

The use of computers in architecture and construction engineering simplifies and automatize many manual operations, especially in the case of curved surfaces such as shell structures. Moreover, it allows fast screening and characterization of many technical solutions through computer-solving equations and the verification of buildings and metallic structures stabilities in different weather and seismic conditions. In parallel, significant efforts have been made to characterize and explore carbon-based nanosystems. Important mathematical concepts and methods were developed for the description of such structures in the frame of mathematical chemistry. Because the lattice topology of shell structures in architecture and nanosystems in chemistry are similar, it is possible to transfer well-established theoretical concepts and knowledge of using nanosystems to the design of shell structures. The topologies of the nanosystems are characterized by lower densities of edges per node offering better solutions for curved surfaces than the typical grids used in architecture. As far as we know, no such connections between the topologies of nanosystems and shell structures have been established before. This transfer would be helpful for increased accuracy and speed in finding the best technical solutions for the building’s design. In this paper, we identify and propose for the design of the shell structures several mathematical approaches developed for atomistic systems.

On ve-Degree Irregularity Index of Graphs and Its Applications as Molecular Descriptor

Most of the molecular graphs in the area of mathematical chemistry are irregular. Therefore, irregularity measure is a crucial parameter in chemical graph theory. One such measure that has recently been proposed is the ve-degree irregularity index (irrve). Quantitative structure property relationship (QSPR) analysis explores the capability of an index to model numerous properties of molecules. We investigate the usefulness of the irrve index in predicting different physico-chemical properties by carrying out QSPR analysis. It is established that the irrve index is efficient to explain the acentric factor and boiling point of molecules with powerful accuracy. An upper bound of irrve for the class of all trees is computed with identifying extremal graphs. We noticed that the result is not correct. In this report, we provide a counter example to justify our argument and determine the correct outcome.

An investigation of edge F-index on fuzzy graphs and application in molecular chemistry

The molecular descriptors are a useful tool in the spectral graph, molecular chemistry and several fields of chemistry and mathematics. The edge F-index is proposed for fuzzy graphs (FGs) here. Bounds of this index are calculated for FGs. The FG has been investigated for a given set of vertices as having maximum edge F-index. Some relations of this index with the second Zagreb index and hyper-Zagreb index are established. For an isomorphic FGs, it is shown that the value of this index is the same. Bounds of this index for some FG operations are determined. Also, an application of the index in mathematical chemistry is studied. For this, 18 octane isomers and 67 alkanes are considered and analyzed the correlation between this index with some properties of the octane isomers and alkanes. From the correlation coefficient value, we have obtained this index is highly correlated with enthalpy of vaporization, standard enthalpy of vaporization, entropy, acentric factor and heat of vaporization and less correlated with heat capacity for octane isomers. Also, this index is correlated with critical pressure, critical temperature, molar refraction, molar volume and boiling point and is less correlated with surface tension and heat of vaporization for alkanes. But, this index is inadequate to determine the melting point of alkanes.

Topological Co-Indices of Molecular Structure of Porphyrazine Network

Topological co-indices are used to measure the strength of a relationship between molecular substance and its properties. A topological co-index is a mathematical representation of a molecular structure that integrates specific physical features of the basic molecular structure and its mathematical representation. Mathematical chemistry uses molecular graphs, which include edges that represent bonds between molecules and vertices that represent atoms, to integrate graph theory with science. Topological co-indices are a type of topological indices that considers separate vertices configurations that are numerical values of the atomic structures, which are used to describe and analyze the various chemical structures. Chemical graph theory brings up more possibilities for scientists. Topological co-indices are frequently used because through this we can save time by determining the topological indices instead of doing time-consuming tests in a research laboratory. In this article, we computed different topological co-indices for the chemical structure of Porphyrazine Network MCIPz(n).

Extremal Harary Index of Graphs with Given Number of Vertices of Odd Degree

The Harary index is one of the most investigated topological indices in mathematical chemistry and in related disciplines, which can be used to describe the molecular structure and physical properties. Let G n , 2 k be the set of all connected graphs on n vertices with exactly 2 k vertices of odd degree. Recently, the graphs in G n , 0 with the first four maximal Harary index and the first minimal Harary index are determined. In the paper, by using the edge-deleted method and some combinatorial skills, we characterize the graph in G n , 2 k with the maximal Harary index for all 0 ≤ k ≤ n / 2 , the graphs in G n , 2 with the first two minimal Harary index, and the graph in G n , 4 with the minimal Harary index. These results will be conducive to the further study of Harary index and its applications.

A unified fixed point approach to study the existence of solutions for a class of fractional boundary value problems arising in a chemical graph theory.

A theory of chemical graphs is a part of mathematical chemistry concerned with the effects of connectedness in chemical graphs. Several researchers have studied the solutions of fractional differential equations using the concept of star graphs. They employed star graphs because their technique requires a central node with links to adjacent vertices but no edges between nodes. The purpose of this paper is to extend the method’s range by introducing the concept of an octane graph, which is an essential organic compound having the formula C8H18. In this manner, we analyze a graph with vertices annotated by 0 or 1, which is influenced by the structure of the chemical substance octane, and formulate a fractional boundary value problem on each of the graph’s edges. We use the Schaefer and Krasnoselskii fixed point theorems to investigate the existence of solutions to the presented boundary value problems in the framework of the Caputo fractional derivative. Finally, two examples are provided to highlight the importance of our results in this area of study.

A Relation on Trees and the Topological Indices Based on Subgraph

A topological index reflects the physical, chemical and structural properties of a molecule, and its study has an important role in molecular topology, chemical graph theory and mathematical chemistry. It is a natural problem to characterize non-isomorphic graphs with the same topological index value. By introducing a relation on trees with respect to edge division vectors, denoted by $\langle\mathcal{T}_n, \preceq \rangle$, in this paper we give some results for the relation order in $\langle\mathcal{T}_n, \preceq \rangle$, it allows us to compare the size of the topological index value without relying on the specific forms of them, and naturally we can determine which trees have the same topological index value. Based on these results we characterize some classes of trees that are uniquely determined by their edge division vectors and construct infinite classes of non-isomorphic trees with the same topological index value, particularly such trees of order no more than $10$ are completely determined.

Computation of Neighborhood M-Polynomial of Three Classes of Polycyclic Aromatic Hydrocarbons

Chemical graph theory is a branch of mathematical chemistry that significantly affects the development of the chemical sciences. The neighborhood M-polynomial approach is an extraordinary process for determining neighborhood degree sum-based topological indices to estimate materials’ various physical, chemical, and biological characteristics. The present work deals with obtaining the NM-polynomial of three types of polycyclic aromatic hydrocarbons (PAHs). Using NM-polynomials as tools, specific degree-based topological indices are retrieved. The applications of these PAHs are pharmaceuticals, lubricating materials, photographic products, agricultural products and other chemical industries. The graphical representations of the outcomes are displayed. These results are compared to several well-known degree-based indices.

Computing the Number of Matchings in Catacondensed Benzenoid Systems

The Hosoya index of is defined as the total number of independent edge sets (number of -matchings ) in . The Hosoya index is one of the most important topological indices in the field of mathematical chemistry because of its relationship with several thermodynamic properties. Therefore, computation of the number of -matchings of various molecular structures has importance. Two methods, one for computing the number of the Hosoya index of catacondensed benzenoid systems and the other for the number of -matchings in benzenoid chains (unbranched catacondensed benzenoid systems), have been presented so far. In this paper, a method based on some transfer matrices to compute the number of -matchings of arbitrary (both unbranched and branched) catacondensed benzenoid systems is presented. Moreover, some algorithms are designed to keep the applicability of the method the same as increases.

On Sum Degree-Based Topological Indices of Some Connected Graphs

In mathematical chemistry, molecular structure of any chemical substance can be expressed by a numeric number or polynomial or sequence of number which represents the whole graph is called topological index. An important branch of graph theory is the chemical graph theory. As a consequence of their worldwide uses, chemical networks have inspired researchers since their development. Determination of the expressions for topological indices of different derived graphs of graphs is a new and interesting problem in graph theory. In this article, some graphs which are derived from honeycomb structure are studied and obtained their exact results for sum degree-based indices. Additionally, a comparison is shown graphically among all the indices.

Analysis of Bismuth (III) Iodide and Dendrimers in Drug Applications

Dendrimers are spherical three-dimensional molecules with a repetitively branching core. They are normally symmetric around the core. Bismuth (III) iodide has the formula B i I 3 and is an inorganic chemical. The reaction between bismuth and iodine produces this gray-black solid, which was of great interest in qualitative inorganic analysis. Mathematical chemistry is an area of mathematics that employs mathematical methods to tackle chemical-related problems. One of these tools is a graphical representation of chemical molecules, known as the molecular graph of a chemical substance. A topological index (TI) is a mathematical function that assigns a numerical value to a (molecular) graph and predicts many physical, chemical, biological, thermodynamical, and structural features of that network. In this work, we will calculate a new topological index, namely, Sombor index, multiplicative Sombor index, and its reduced version for bismuth (III) iodide and dendrimers. We also plot our computed results of Sombor index, multiplicative Sombor index, and reduced Sombor index to examine how they were affected by the parameters involved.

ADMET and quantitative structure property relationship analysis of anti‐Covid drugs against omicron variant with some degree‐based topological indices

AbstractChemical graph theory is one of the important fields in mathematical chemistry which provides a useful tool called topological indices which transforms chemical structure of a molecule into a numerical value. Topological indices are used to investigate physicochemical properties, pharmaco‐kinetic properties, biological activity in quantitative structure property relationship (QSPR) and quantitative structure–activity relationship. The most life‐threatening disease the world facing is COVID‐19 and its prominent variants like alpha, beta, delta and omicron variants. The delta and omicron variants are deemed one of the most contagious strains of the SARS‐CoV‐2 virus. Since there is no exact drug for these variants, but researches are been carried out with some existing and new drugs which are effective against delta and omicron variants. Favipiravir, Baricitinib, Fluvoxamine, Nirmatrelvir and Molnupiravir are some of the drugs currently used to treat COVID‐19 omicron variant. In this paper QSPR model is designed to predict some selected physicochemical properties and ADMET properties of the above aforesaid drugs by using some degree‐based indices such as first K Banhatti index, second K Banhatti index, first K hyper Banhatti index, second K hyper Banhatti index, modified first K Banhatti index, modified second K Banhatti index and harmonic K Banhatti index via M‐Polynomial. The relationship analyses for the physicochemical and ADMET properties with the degree‐based indices are done by using the quadratic and cubic regression method respectively. The results obtained can be expanded to correlate several other properties associated with drugability of potential candidates that can further be utilized to construct a disease‐based library of drugs.

A Relation Between Wiener Index and Mostar Index for Daisy Cubes

Daisy cubes are a class of isometric subgraphs of the hypercubes Q n. Daisy cubes include some previously well known families of graphs like Fibonacci cubes and Lucas cubes. Moreover they appear in chemical graph theory. Two distance invariants, Wiener and Mostar indices, have been introduced in the context of the mathematical chemistry. The Wiener index W (G) is the sum of distance between over all unordered pairs of vertices of a graph G. The Mostar index Mo(G) is a measure of how far G is from being distance balanced. In this paper we establish that the Wiener and the Mostar indices of a daisy cube G are linked by the relation 2W (G) -- Mo(G) = |V (G)||E(G)|. We give also an expression of Wiener and Mostar indices for daisy cubes.

Computational and comparative aspects of two carbon nanosheets with respect to some novel topological indices

The recent discovery of various forms of carbon nanostructures inspired the researchers due to their uses in various fields. Numerous production approaches for carbon nanostructures have been presented; doping, functionalization, chemical modification, and filling have been achieved, and manipulation, characterization and separation of carbon nanostructures are now possible. On the other hand, the flexibility and strength of carbon nanosheets make them potential candidates in controlling further nanoscale structures, which indicates that they have an important role in nanotechnology engineering. Because of their uniform thickness of only around 1 nanometer( nm) , and their high mechanical, thermal and chemical stability, such materials are of high concern for a wide range of applications. A topological index or molecular descriptor is a mathematical formulation that can be applied to any graph which models some molecular structure. Based on the molecular descriptor, it is convenient to analyze mathematical values and advance investigation on some physicochemical properties of a molecule. Consequently, it is an effective method in replacing laborious, expensive and time-consuming laboratory experimentations. Molecular descriptors perform an important part in mathematical chemistry, especially during the investigation of Quantitative structure property relationships (QSPRs) and Quantitative structure-activity relationships (QSARs) that provide insight into animate effects based on chemical structures. In this paper, we have computed some novel neighborhood version molecular descriptors for the two carbon nanosheets V C 5 C 7 p , q and ( H C 5 C 7 p , q ) and derived formulas for them. Our computed results are surely helpful in understanding the topology of understudy nanosheets. These computed indices have best correlation with acentric factor and entropy, therefore, are effective in QSPRs and QSARs analysis with powerful accuracy.

Total Irregularity Strengths of a Disjoint Union of (4,6)-Fullerene Graphs

A fullerene is a polyhedral molecule made entirely of carbon atoms other than graphite and diamond. Mathematical chemistry or Chemical graph theory as a combination of chemistry and graph theory studies the physical properties of mathematically modeled chemical compounds. Graph labeling is a vital area of graph theory that has applications in both physical and social sciences. For example, in chemistry vertex labeling is used in the constitution of valence isomers and transition labeling to study chemical reaction networks. A (4,6)-fullerene graph is a mathematical model of a (4,6)-fullerene molecule. It is a cubic-regular planar graph whose faces are squares and hexagons. In this article, we study the total vertex irregularity strength and the total edge irregularity strength of a disjoint union of (4,6)-fullerene graphs and provide their exact values.

Eccentricity-Based Topological Descriptors of First Type of Hex-Derived Network

Graph theory has made significant progress in mathematical chemistry and has gained a lot of traction among scientists due to its numerous applications in mathematical chemistry. The numerical invariants of a molecular structure are known as molecular topological descriptors, and they are highly effective for predicting their bioactivity. A number of such indices are examined and applied in pharmaceutical research, chemistry, medication development, and other fields. The eccentricity-based Zagreb indices, total eccentricity, geometric arithmetic G A 4 , atom-bond connectivity A B C 5 , and average eccentricity indices of a hex-derived network of first type are computed in this article. We also provide analytically closed formulas for these descriptors, which may be used to investigate the underlying topologies.

SIGEST

The SIGEST article in this issue is Hamiltonicity of Cubic Planar Graphs with Bounded Face Sizes, by František Kardoš. The original version of this article appeared in the SIAM Journal on Discrete Mathematics in 2020. Barnette's Conjecture is a famous problem in graph theory which asserts that every cubic bipartite 3-connected planar graph has a Hamiltonian cycle. The conjecture stems from an attempt in 1880 by P. G. Tait to solve the Four Color Theorem by establishing that every cubic 3-connected planar graph has a Hamiltonian cycle. This is a stronger statement than the Four Color Theorem and it actually turns out to be false, as shown by W. T. Tutte in 1946. Barnette's Conjecture is equivalent to the statement that every plane cubic 3-connected graph with all faces of even size has a Hamiltonian cycle. Our SIGEST paper proves a closely related conjecture of Barnette and Goodey asserting that every plane cubic 3-connected graph with all faces of size at most six has a Hamiltonian cycle; this also solves an open problem from mathematical chemistry, which posited the existence of a Hamiltonian cycle under the stronger assumption that all faces have size five or six. The proof consists of several reduction steps and eventually boils the problem down to checking a finite number of configurations, which is handled with the assistance of a computer. The derivations are accompanied by high-quality graphical illustrations, and the author has provided computer code for the automated parts of the proof.

Hubungan Kemampuan Matematika dan Kemampuan Fisika Terhadap Hasil Belajar Siswa Pada Materi Termokimia

This research was conducted to determine whether there is a significant relationship between mathematical skills, physics skills on students' chemistry learning outcomes, as well as the relationship between mathematical skills and physics skills on chemistry learning outcomes. The sample in this study consisted of one class which was taken using a random sampling technique (lottery). The instrument used was a test instrument which consisted of 3 types, namely a math skills test, a physics skills test and a chemistry learning outcome test. Data analysis was performed using simple regression and multiple regression, as well as the correlation test and the coefficient of determination at the significant level α = 0.05. The results showed that there was a significant relationship between mathematical skills and students' chemistry learning outcomes, there was a significant relationship between physics skills and students' chemistry learning outcomes, then it was examined that there was a significant relationship between mathematical skills and physics skills on students' chemistry learning outcomes. The contribution of mathematical skills and physics skills to student learning outcomes was 74.30%, meaning that 25.70% was the contribution of other factors.

Analisis Hubungan Antara Kemampuan Matematika dan Analisis Kimia Terhadap Hasil Belajar Kimia Materi Kesetimbangan Kimia

This research was conducted at SMA Negeri 7 Medan on the Analysis of the Relationship Between Mathematical Ability and Chemical Analysis of Students' Chemistry Learning Outcomes in Chemical Equilibrium Material. The population in this study were all students of class XI MIPA SMA Negeri 7 Medan as many as 7 classes. Sampling using purposive sampling technique. The sample taken in this study was class XI MIPA 2. The instruments used consisted of: (1) mathematical ability test instruments in the form of multiple choice questions; (2) chemical analysis ability test instrument in the form of essay questions; (3) the test instrument for chemistry learning outcomes is in the form of multiple choice questions. The results of this study are: (1) there is a linear and significant relationship between mathematical ability and chemistry learning outcomes with a value of Sig(1-tailed) 0.000 <0.05; (2) there is a linear and significant relationship between analytical ability on chemistry learning outcomes with a Sig(1-tailed) value of 0.000 <0.05; (3) there is a significant relationship between mathematical ability and chemical analysis ability on chemistry learning outcomes with a value of Sig(1-tailed) 0.000<0.05.