Narrow Results

Modeling and computation of molecular descriptors play a crucial role in understanding the properties and characteristics of novel materials like $\gamma$-graphyne, the next-generation wonder material. Triangular $\gamma$-graphyne is an intriguing two-dimensional carbon-based material with diverse potential applications due to its unique electronic, mechanical, and thermal properties. Triangular $\gamma$-graphyne, a distinctive form of carbon allotrope, has garnered substantial interest as a result of its extraordinary characteristics. The notable value lies in its ability to be utilized in diverse domains for potential applications. The interconnected mesh structure of Triangular $\gamma$-graphyne exhibits exceptional mechanical strength and exceptional electron-transporting properties. This makes it a promising material for the development of high-performance electronic devices, such as transistors and sensors. Furthermore, gamma graphyne possesses excellent thermal stability, which makes it suitable for applications in the field of energy storage and conversion. It has the potential to be used as a catalyst in fuel cells or as an electrode material in supercapacitors. In this study, we utilize graph theory to dissect the molecular structure of triangular $\gamma$-graphyne, leading to the derivation of specific mathematical formulas for essential degree-based molecular characteristics. These findings can be instrumental in examining the correlations between the structure and properties of $\gamma$-graphyne. This paper focuses on two structures made from hexagonal honeycomb graphite lattices, like triangular $\gamma$-graphyne and triangular $\gamma$-graphyne chains with respect to some degree-based topological indices. The results obtained will aid in the investigation of the structure-property relationships in $\gamma$-graphyne.

Quantitative structure property relationship (QSPR) is essential in rational drug design by facilitating the prediction, optimization, and prioritization of potential drug candidates based on their chemical structures and properties, optimizing the utilization of computational resources and minimizing costs. Topological indices play a fundamental role in QSPR studies by providing a quantitative representation of molecular structures and facilitating the prediction of various chemical properties and activities. Metastatic skin cancer can be aggressive and potentially fatal, and thus developing effective drugs can improve patient outcomes, including overall survival rates and progression-free survival. This research work focuses on the structural behaviors of medications used in the treatment of skin cancer, including binimetinib, fisetin, encorafenib, picato, fluorouracil, trametinib, vemurafenib, imiquimod, odomzo, vismodegib, dacarbazine, cobimetinib, dabrafenib, sesamol, curcumin, doxorubicin, temozolomide, paclitaxel, itraconazole, and hyaluronan. We have developed QSPR models involving degree and neighborhood degree sum-based indices and conducted a comparative analysis to highlight the efficacy of the models.

Molecular descriptors, such as topological indices (TIs), play a crucial role in network theory, spectral graph theory, and molecular chemistry. Spherical fuzzy graphs (SFGs), an extension of picture fuzzy graphs (PFGs), utilize topological indices from crisp graphs to provide a broader analytical framework. This article defines second Zagreb, Randić, arithmetic-geometric, geometric-arithmetic, and Sombor indices and matrices for SFGs, along with their energy and Laplacian measures. An example demonstrates how SFGs energies relate to these indices. Lastly, SFGs aid decision-making by evaluating these energies in comparative analyses to determine optimal links between neighboring countries.

Graph theory has many applications in chemistry and is used to analyze molecular structures. Topological descriptors are numerical numbers that contain chemical information and provide structural features of a compound associated with a chemical approach. The purpose of the topological index is to study the physicochemical properties of molecular structures. This paper investigates the molecular graph of 2D silicon carbide structures. The scope of this paper is to determine the highest thermal stability property among silicon carbide structures using topological indices.

The purpose of this paper is to discuss the use of topological indices (TIs) to anticipate the physical and biological aspects of innovative drugs used in the treatment of Alzheimer’s disease. Degree-based topological indices are generated using edge partitioning to assess the drugs Tacrine, Donepezil, Ravistigmine, Butein, Licochalcone-A and Flavokqwain-A. Furthermore, using linear regression, a quantitative structure–property relationship (QSPR) model is developed to predict the characteristics such as boiling point (BP), flash point (FP), molar volume (MV), molecular weight, complexity and polarizability. The findings show that topological indices have the potential to be used as a tool for drugs discovery and design in the field of Alzheimer’s disease treatment.

The sequence and structure of a large body of proteins are becoming increasingly available. It is desirable to explore mathematical tools for efficient extraction of information from such sources. The principles of graph theory, which was earlier applied in fields such as electrical engineering and computer networks are now being adopted to investigate protein structure, folding, stability, function and dynamics. This review deals with a brief account of relevant graphs and graph theoretic concepts. The concepts of protein graph construction are discussed. The manner in which graphs are analyzed and parameters relevant to protein structure are extracted, are explained. The structural and biological information derived from protein structures using these methods is presented.

The relationship between the topological indices and the Neutral Endopeptidase (NEP) inhibitory activity and Angiotensin-Converting Enzyme (ACE) inhibitory activity of mercaptoacyldipeptides has been investigated. Three topological indices — the Wiener index (a distance-based topological index), the molecular connectivity index (an adjacency-based topological index), and the eccentric connectivity index (an adjacency-cum-distance-based topological index), were presently used for investigation. A data set comprising 39 differently substituted mercaptoacyldipeptides was selected for the present study. The values of the Wiener index, molecular connectivity index, and eccentric connectivity index for each of the 39 compounds comprising the data set were computed using an in-house computer program. Resultant data were analyzed and suitable models were developed after identification of the active ranges. Subsequently, a biological activity was assigned to each compound using these models, and the biological activity was then compared with the reported NEP and ACE inhibitory activity of each compound. Accuracy of prediction up to a maximum of ~91% was obtained using these models.

Transition-metal-catalyzed homo and hetero coupling is a rapidly growing area of research. This work refers to the nickel-catalyzed photoinduced amination study of $\beta$-bromotetraarylporphyrin and its Ni(II), Zn(II) and Cu(II) complexes. The selective mono $\beta$-bromination of 5,10,15,20-tetrakis(4$\prime$-isopropylphenyl)porphyrin was achieved with $N$-bromosuccinimide. Under similar conditions, $\beta$-bromination of Ni(II), Zn(II) and Cu(II) complexes of 5,10,15,20-tetrakis(4$\prime$-isopropylphenyl)porphyrin successfully afforded the corresponding 2-bromometalloporphyrins. The $\beta$-bromoporphyrin/metalloporphyrins were coupled with three different amines through the creation of the C–N bond by using an economical and air-tolerant photoactive catalyst (NiBr${}_2^{}$ · 3H₂O) at room temperature under 365 nm radiations. Nickel-catalyzed amination yields are compared with the traditional Buchwald–Hartwig amination yields. Due to the low operational cost, photoinduced nickel-catalyzed C–N couplings were found to be more economical than the Buchwald–Hartwig amination procedure, although the latter afforded higher yields. The nickel-catalyzed photoamination reaction was also extended for the one-pot synthesis of pyridine-3,5-diamine bridged porphyrin dyad. The intramolecular cyclization of the pyridine-3,5-diamine-bridged porphyrin dyad afforded a novel quinolino-fused porphyrin dyad. Degree- and distance-based topological indices of the newly synthesized porphyrins were calculated and correlated with their molar refractivity. All newly synthesized porphyrins are characterized by UV-vis, FTIR, ¹H NMR, elemental analysis and mass spectrometry.

Let $G$ be a simple connected graph with $n$ vertices, $m$ edges and let ${\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_{n-1}>{\mu }_n=0$ be its Laplacian eigenvalues. The Laplacian resolvent energy of graph $G$ is defined by $\mathrm{\text{RL}}(G)={\sum }_{i=1}^n\frac{1}{n+1-{\mu }_i}$. In this paper, we give some new lower bounds for the invariant $\mathrm{\text{RL}}(G)$.

In a connected graph $G$, the distance between two vertices of $G$ is the length of a shortest path between these vertices. The eccentricity of a vertex $u$ in $G$ is the largest distance between $u$ and any other vertex of $G$. The total-eccentricity index $\tau (G)$ is the sum of eccentricities of all vertices of $G$. In this paper, we find extremal trees, unicyclic and bicyclic graphs with respect to total-eccentricity index. Moreover, we find extremal conjugated trees with respect to total-eccentricity index.

Graphene is an atomic scale honeycomb lattice made of the carbon atoms. Graph theory has given scientific expert an assortment of helpful apparatuses, for example, topological indices. A topological index $\mathrm{\text{Top}}(G)$ of a graph $G$ is a number with the property that for each graph $H$ isomorphic to $G,$$\mathrm{\text{Top}}(H)=\mathrm{\text{Top}}(G).$ In this paper, we exhibit correct expressions for some topological indices for para-line graph of honeycomb networks and graphene.

Let $I(G)$ be a topological index of a graph. If $I(G+e)<I(G)$ (or $I(G+e)>I(G)$, respectively) for each edge $e\notin G$, then $I(G)$ is monotonically decreasing (or increasing, respectively) with the addition of edges. In this paper, by a unified approach, we determine the extremal values of some monotonic topological indices, including the Wiener index, the hyper-Wiener index, the Harary index, the connective eccentricity index, the eccentricity distance sum, among all connected bipartite graphs with a given number of cut edges, and characterize the corresponding extremal graphs, respectively.

Graph operations play a significant role in constructing new and valuable graphs and capturing intermolecular forces between atoms and bonds of a molecule. In mathematical chemistry and chemical graph theory, a topological invariant is a numeric value extracted from the molecular graph of a chemical compound using a mathematical formula involving vertex degrees, distance, spectrum, and their combination. An intriguing problem in chemical graph theory is figuring out the lower and the upper bound on pertinent topological indices among a particular family of graphs. The first Gourava index for a graph $\mathrm{\Gamma }$ is denoted and defined as ${\mathrm{\text{GO}}}_1(\mathrm{\Gamma })={\sum }_{xy\in E(\mathrm{\Gamma })}[{𝔡}_{\mathrm{\Gamma }}(x)+{𝔡}_{\mathrm{\Gamma }}(y)+{𝔡}_{\mathrm{\Gamma }}(x){𝔡}_{\mathrm{\Gamma }}(y)].$ Recently, Kulli studied and derived formulas of the first Gourava index for four graph operations. We proved with the help of counter-examples that the results provided by Kulli produce inaccurate values when compared with exact values. In this paper, we determined the exact formulas and bounds of the first Gourava index for $\mathrm{\Psi }$-sum graphs. Besides, we presented diverse examples to support our results.

In this study, we suggest an eccentricity-based linear regression model to predict the effectiveness of anti-Alzheimer’s drugs such as Tacrine, Donepezil, Rivastigmine, Butein, Licochalcone-A and Flavokawain-A. The relationship between structural characteristics and therapeutic effectiveness by integrating eccentricity values is derived from these drugs. Our findings provide promising insights into the potential use of eccentricity-driven linear regression as a drug discovery tool in the field of Alzheimer’s disease treatment. This method provides an improved comprehension of the structural factors influencing drug efficacy, establishing the way for more targeted and efficient anti-Alzheimer’s drug development. Moreover, a quantitative structure–property relationship model is developed by using linear regression model to predict properties such as boiling point, flash point, molar volume, molecular weight, complexity and polarizability.

Polycyclic aromatic hydrocarbons (PAHs) have distinctive chemical structures and are well known for their wide range of uses and environmental relevance. This work explores the impact of these structural characteristics on eccentricity-based topological indices offering information about the arrangement of atoms within the molecules. This study uses quantitative structure-property relationship (QSPR) analysis to construct prediction models for understanding and forecasting specific PAHs including Dibenzo[e,l]Pyrene, Heptacence, Heptaphene, Naphthacene, Naphthalene, Naphto[1,2a]pyrene, Naphtho[2,3a]pyrene, perylene, Perylene, Picene, Phenanthrene, Pyrene, Tetraphene, Tribenzo[b,n,pqr]perylene, Tribenzo[a,fg,op]tetracene and Triphenylene. Furthermore, regression analysis applies to clarify the quantitative correlations between the factors under study and improves the interpretability of the data produced. The combined use of these diverse approaches advances a thorough comprehension of the mutual influence of chemical structure, topological indices and predictive modeling about PAHs.

Mathematical Chemistry is concerned with the use of mathematics to solve problems in Chemistry. In Chemistry, the molecules are frequently shown as graphs with vertices denoting atoms and edges denoting bonds, respectively. Atom valences and bond multiplicities are represented by vertex degrees and edge multiplicities, respectively. The vertices in a graph are said to be close by if an edge connects them. Alkanes are a class of compounds whose physical characteristics are modelled using the chemical graph theory. The melting and boiling points of the molecules are modelled using topological indices based on the graphical structure of the alkanes. A mesh network is a local area network, in which the infrastructure nodes (i.e., bridges, switches and other infrastructure devices) connect directly, dynamically and non-hierarchically to as many other nodes as possible and cooperate with one another to efficiently route data. In this paper, the distinct degrees of triangular mesh network, enhanced mesh network, rhenium tri-oxide lattice network and star of silicate network are list with edge partitions technique. We determine the first and second Zagreb index, the modified second Zagreb index, the symmetric division index, the harmonic index and the inverse sum index. We characterise detailed proofs of these indices for each triangular network. Furthermore, their graphical representation and comparison between the certain topological indices is presented in Sec. 6. These indices are highly accurate in the study of QSPRs and QSARs because they have strong correlation with the acentric factor and entropy.

This study explores the utilization of topological graph invariants, or molecular descriptors, to mathematically model chemical compounds. These descriptors are vital in quantifying the physio-chemical characteristics of a compound, and are commonly represented as shapes such as polygons, bushes, and grapes. Specifically, this research focuses on computing selected fifth multiplicative first and second Zagreb indices, third and fourth multiplicative general fifth multiplicative Zagreb indices, and other degree-based topological indices for the benzene ring graph (${\mathrm{\text{BR}}}_P)$ and the simple bounded dual of benzene ring graph (${\mathrm{\text{SBR}}}_P)$. The findings of this study are then compared to demonstrate the impact of these molecular descriptors.

Topological indices are numeric quantities that transform chemical structure to real number. Topological indices are used in QSAR/QSPR studies to correlate the bioactivity and physiochemical properties of molecule. In this paper, some newly designed neighborhood degree-based topological indices named as neighborhood Zagreb index ($M_N$), neighborhood version of Forgotten topological index ($F_N$), modified neighborhood version of Forgotten topological index ($F_N^{\ast }$), neighborhood version of second Zagreb index ($M_2^{\ast }$) and neighborhood version of hyper Zagreb index ($H{M}_N$) are obtained for Graphene and line graph of Graphene using subdivision idea. In addition, these indices are compared graphically with respect to their response for Graphene and line graph of subdivision of Graphene.