Molecular Graph Research Articles

Counting Rules for Computing the Number of Independent Sets of a Grid Graph

The issue of counting independent sets of a graph, G, represented as i(G), is a significant challenge within combinatorial mathematics. This problem finds practical applications across various fields, including mathematics, computer science, physics, and chemistry. In chemistry, i(G) is recognized as the Merrifield–Simmons (M-S) index for molecular graphs, which is one of the most relevant topological indices related to the boiling point in chemical compounds. This article introduces an innovative algorithm designed for tallying independent sets within grid-like structures. The proposed algorithm is based on the ‘branch-and-bound’ technique and is applied to compute i(Gm,n) for a square grid formed by m rows and n columns. The proposed approach incorporates the widely recognized vertex reduction rule as the basis for splitting the current subgraph. The methodology involves breaking down the initial grid iteratively until outerplanar graphs are achieved, serving as the ’basic cases’ linked to the leaf nodes of the computation tree or when no neighborhood is incident to a minimum of five rectangular internal faces. The time complexity of the branch-and-bound algorithm speeds up the computation of i(Gm,n) compared to traditional methods, like the transfer matrix method. Furthermore, the scope of the proposed algorithm is more general than the algorithms focused on grids since it could be applied to process general mesh graphs.

Quantitative structural elucidation of original lignins in different varieties of waste Gleditsia sinensis Lam. pods

Obtaining new materials, industrial chemicals, and energy from the conversion of agricultural and forestry biomass to fill the gap of non-renewable fossil energy has become a growing trend. As a waste biomass, the G. sinensis pods (GSP) after saponin extraction are always discarded due to lack of industrial/high value-added utilization. Herein, in order to lay a foundation for the selection of pretreatment strategies of GSP and the valorization of GSP lignin, the lignins from the different varieties of GSP were isolated by double enzymatic method and quantitatively characterized. Results indicated that GSP lignin was of the GS-type (S/G, 1.1–1.4) and the interunit linkages were mainly β-O-4′ linkage, along with a certain amount of carbon-carbon linkages (≈35/Ar). L1 showed higher β-O-4′ linkage content (61.3/100 Ar) and relative molar mass (12190 mol/g). Some atypical lignin units, such as cinnamyl aldehyde end-group, cinnamyl alcohol end-groups, and ferulates, were also revealed in GSP lignins. Finally, the potential molecular structure graphs of the GSP lignin were depicted on the basis of the quantitative results. Overall, comprehensive research of the structural characteristics of GSP lignin is essential for efficiently deconstructing GSP and transforming the utilization of the cell wall components.

Sufficient conditions for [formula omitted]-leaf-connected graphs in terms of the first Zagreb index, the reciprocal degree distance and the forgotten topological index

For integer k≥2, a graph G is called k-leaf-connected if |V(G)|≥k+1 and given any subset S⊆V(G) with |S|=k, G always has a spanning tree T such that S is precisely the set of leaves of T. Thus a graph is 2-leaf-connected if and only if it is Hamiltonian-connected. The first Zagreb index M1(G) is equal to the sum of squares of the degrees of the vertices of the underlying molecular graph G. The reciprocal degree distance (RDD), defined as vertex-degree-weighted sum of the reciprocal distances, that is, RDD(G)=∑vi≠vjdegG(vi)+degG(vj)dG(vi,vj), where degG(vi) is the degree of the vertex vi in G and dG(vi,vj) denotes the distance between two vertices vi and vj in G. The forgotten topological index of G is the sum of cubes of all its vertex degrees, which plays a significant role in measuring the branching of the carbon atom skeleton. Finding sufficient conditions for graphs possessing certain properties is an important and meaningful problem. In this paper, we give sufficient conditions in terms of the first Zagreb index, the reciprocal degree distance or the forgotten topological index for a graph to be k-leaf-connected, which generalizes two results of An (2022). All of these results are best possible.

A QSPR analysis of physical properties of antituberculosis drugs using neighbourhood degree-based topological indices and support vector regression

Topological indices are molecular descriptors used in QSPR modelling to predict the physicochemical properties of molecules. Topological indices are used in numerous applications in drug design. In this work, we compute the neighbourhood degree-based topological indices of 15 antituberculosis drugs, we studied the QSPR analysis of these drugs using support vector regression. The efficiency of support vector regression is determined by comparing it with the classical linear regression. Our QSPR model further shows the superiority of the SVR model as a better predictive model in QSPR analysis of the physical properties of antituberculosis drugs. The findings in this study are a further contribution to the field of chemical graph theory and drug design, providing a deeper understanding of neighbourhood degree-based topological indices and their predictive capabilities in QSPR model.

Multimodal Transformer for Property Prediction in Polymers.

In this work, we designed a multimodal transformer that combines both the Simplified Molecular Input Line Entry System (SMILES) and molecular graph representations to enhance the prediction of polymer properties. Three models with different embeddings (SMILES, SMILES + monomer, and SMILES + dimer) were employed to assess the performance of incorporating multimodal features into transformer architectures. Fine-tuning results across five properties (i.e., density, glass-transition temperature (Tg), melting temperature (Tm), volume resistivity, and conductivity) demonstrated that the multimodal transformer with both the SMILES and the dimer configuration as inputs outperformed the transformer using only SMILES across all five properties. Furthermore, our model facilitates in-depth analysis by examining attention scores, providing deeper insights into the relationship between the deep learning model and the polymer attributes. We believe that our work, shedding light on the potential of multimodal transformers in predicting polymer properties, paves a new direction for understanding and refining polymer properties.

GraphormerDTI: A graph transformer-based approach for drug-target interaction prediction

The application of Artificial Intelligence (AI) to screen drug molecules with potential therapeutic effects has revolutionized the drug discovery process, with significantly lower economic cost and time consumption than the traditional drug discovery pipeline. With the great power of AI, it is possible to rapidly search the vast chemical space for potential drug-target interactions (DTIs) between candidate drug molecules and disease protein targets. However, only a small proportion of molecules have labelled DTIs, consequently limiting the performance of AI-based drug screening. To solve this problem, a machine learning-based approach with great ability to generalize DTI prediction across molecules is desirable. Many existing machine learning approaches for DTI identification failed to exploit the full information with respect to the topological structures of candidate molecules. To develop a better approach for DTI prediction, we propose GraphormerDTI, which employs the powerful Graph Transformer neural network to model molecular structures. GraphormerDTI embeds molecular graphs into vector-format representations through iterative Transformer-based message passing, which encodes molecules' structural characteristics by node centrality encoding, node spatial encoding and edge encoding. With a strong structural inductive bias, the proposed GraphormerDTI approach can effectively infer informative representations for out-of-sample molecules and as such, it is capable of predicting DTIs across molecules with an exceptional performance. GraphormerDTI integrates the Graph Transformer neural network with a 1-dimensional Convolutional Neural Network (1D-CNN) to extract the drugs' and target proteins' representations and leverages an attention mechanism to model the interactions between them. To examine GraphormerDTI's performance for DTI prediction, we conduct experiments on three benchmark datasets, where GraphormerDTI achieves a superior performance than five state-of-the-art baselines for out-of-molecule DTI prediction, including GNN-CPI, GNN-PT, DeepEmbedding-DTI, MolTrans and HyperAttentionDTI, and is on a par with the best baseline for transductive DTI prediction. The source codes and datasets are publicly accessible at https://github.com/mengmeng34/GraphormerDTI.

Exploring Chemical Reaction Space with Machine Learning Models: Representation and Feature Perspective.

Chemical reactions serve as foundational building blocks for organic chemistry and drug design. In the era of large AI models, data-driven approaches have emerged to innovate the design of novel reactions, optimize existing ones for higher yields, and discover new pathways for synthesizing chemical structures comprehensively. To effectively address these challenges with machine learning models, it is imperative to derive robust and informative representations or engage in feature engineering using extensive data sets of reactions. This work aims to provide a comprehensive review of established reaction featurization approaches, offering insights into the selection of representations and the design of features for a wide array of tasks. The advantages and limitations of employing SMILES, molecular fingerprints, molecular graphs, and physics-based properties are meticulously elaborated. Solutions to bridge the gap between different representations will also be critically evaluated. Additionally, we introduce a new frontier in chemical reaction pretraining, holding promise as an innovative yet unexplored avenue.

Self-Supervised Contrastive Molecular Representation Learning with a Chemical Synthesis Knowledge Graph.

Self-supervised molecular representation learning has demonstrated great promise in bridging machine learning and chemical science to accelerate the development of new drugs. Due to the limited reaction data, existing methods are mostly pretrained by augmenting the intrinsic topology of molecules without effectively incorporating chemical reaction prior information, which makes them difficult to generalize to chemical reaction-related tasks. To address this issue, we propose ReaKE, a reaction knowledge embedding framework, which formulates chemical reactions as a knowledge graph. Specifically, we constructed a chemical synthesis knowledge graph with reactants and products as nodes and reaction rules as the edges. Based on the knowledge graph, we further proposed novel contrastive learning at both molecule and reaction levels to capture the reaction-related functional group information within and between molecules. Extensive experiments demonstrate the effectiveness of ReaKE compared with state-of-the-art methods on several downstream tasks, including reaction classification, product prediction, and yield prediction.

Molecular descriptor-enhanced graph neural network for energetic molecular property prediction

Molecular descriptor-enhanced graph neural network for energetic molecular property prediction

Investigating Banhatti indices on the molecular graph and the line graph of Glass with M-polynomial approach

Topological indices are numerical values related to a chemical structure that describes the correlation of chemical structure with different physical properties and chemical reactions. Glass has wide applications in architecture, tableware, optics, and optoelectronics. In this article, first, the mathematical relationship between M-polynomial and Banhatti indices such as K-Banhatti, d-Banhatti, and hyper d-Banhatti indices are obtained. Then using M-polynomial, Banhatti indices are calculated.

Progressive Multiple Alignment of Graphs

The comparison of multiple (labeled) graphs with unrelated vertex sets is an important task in diverse areas of applications. Conceptually, it is often closely related to multiple sequence alignments since one aims to determine a correspondence, or more precisely, a multipartite matching between the vertex sets. There, the goal is to match vertices that are similar in terms of labels and local neighborhoods. Alignments of sequences and ordered forests, however, have a second aspect that does not seem to be considered for graph comparison, namely the idea that an alignment is a superobject from which the constituent input objects can be recovered faithfully as well-defined projections. Progressive alignment algorithms are based on the idea of computing multiple alignments as a pairwise alignment of the alignments of two disjoint subsets of the input objects. Our formal framework guarantees that alignments have compositional properties that make alignments of alignments well-defined. The various similarity-based graph matching constructions do not share this property and solve substantially different optimization problems. We demonstrate that optimal multiple graph alignments can be approximated well by means of progressive alignment schemes. The solution of the pairwise alignment problem is reduced formally to computing maximal common induced subgraphs. Similar to the ambiguities arising from consecutive indels, pairwise alignments of graph alignments require the consideration of ambiguous edges that may appear between alignment columns with complementary gap patterns. We report a simple reference implementation in Python/NetworkX intended to serve as starting point for further developments. The computational feasibility of our approach is demonstrated on test sets of small graphs that mimimc in particular applications to molecular graphs.

Predictive potential of eigenvalues-based graphical indices for determining thermodynamic properties of polycyclic aromatic hydrocarbons with applications to polyacenes

For the structure–property modeling of thermodynamic properties of polycyclic aromatic hydrocarbons (PAHs), this paper explores the effectiveness of a specific class of graphical indices defined based on eigenvalues, also known as valency-spectral graphical indices. The 30 lower PAHs have been selected as test molecules. The heat capacity (Cp) and entropy (So) are the standard choices for characterizing thermodynamic properties due to their close correlation with the thermodynamic properties of a similar nature. A computer-dependent method is proposed to calculate valency-spectral indices of chemical graphs, and it is employed to compute the indices for our test molecules. The statistical inferences reveal an unexpected finding for the Estrada harmonic energy, a less familiarized spectral descriptor within the research community, outperforming all the existing descriptors, followed by the positive & negative inertia indices. The Estrada harmonic energy achieved the highest mean correlation coefficient of ρ>0.99. On the other hand, well-studied descriptors such as Laplacian energy and adjacency energy still maintained their effectiveness in correlating the thermodynamic properties of PAHs; however, they were significantly outperformed by the Estrada harmonic energy. The three best valency-spectral indices are employed in the structure–property modeling of thermodynamic properties of linear polyacenes.

Computation of wiener polynomial and index of line subdivision friendship and line subdivision bifriendship graphs using matlab program

A topological index is a branch of chemical graph theory that is vital to analyzing the physio-chemical characteristics of chemical compound structures divided into a degree-based molecular structure such as Zagreb indices, a distance-based molecular structure such as Wiener index, and a mixed such as Gutman index. In this paper, some definitions, results, and examples of Wiener polynomial and index for subdivision graph of friendship, bifriendship graphs, line subdivision graph of friendship, and bifriendship graphs were introduced. Moreover, we used the MATLAB program to calculate the Wiener polynomial and index of these graphs and refer to some applications.

Development and uniqueness test of highly selective atomic topological indices based on the number of attached hydrogen atoms

On the basis of the atomic graph-theoretical index – aEAID (atomic Extended Adjacency matrix IDentification) and molecular adjacent topological index – ATID (Adjacent Topological IDentification) suggested by one of the authors (Zhang Q), a highly selective atomic topological index - aATID (atomic Adjacent Topological IDentification) index was suggested to identify the equivalent atoms in this study. The aATID index of an atom was derived from the number of the attached hydrogen atoms of the atom but omitting bond types. In this case, the suggested index can be used to identify equivalent atoms in chemistry but perhaps not equivalent in the molecular graph. To test the uniqueness of aATID indices, the virtual atomic data sets were derived from alkanes containing 15–20 carbon atoms and the isomers of Octogen, as well as a real data set was derived from the NCI database. Only four pairs of atoms from alkanes containing 20 carbons can't be discriminated by aATID, that is, four pairs of degenerates were found for this data set. To solve this problem, the aATID index was modified by introducing distance factors between atoms, and the 2-aATID index was suggested. Its uniqueness was examined by 5,939,902 atoms derived from alkanes containing 20 carbons and further 16,166,984 atoms from alkanes of 21 carbons, and no degenerates were found. In addition, another large real data set of 16,650,688 atoms derived from the PubChem database was also used to test the uniqueness of both aATID and 2-aATID. As a result, each atom was successfully discriminated by any of the two indices. Finally, the suggested aATID index was applied to the identification of duplicate atoms as data pretreatment for QSPR (Quantitative Structure–Property Relationships) studies.

On the metric-based resolving parameter of the line graph of certain structures

Let G be a graph and R = {r1, r2, …, rk} be an ordered subset of vertices of G, if every two vertices of G have different representation r (v|R) = (d (v, r1) , d (v, r2) , …, d (v, rk)) with respect to R, then R is said to be a metric-based resolving parameter or resolving set of G and its minimum cardinality is called the metric dimension of graph G . Metric dimension is considered as an important applied concept of graph theory especially in the localization of a network and also in the chemical graph theoretical study of molecular compounds. Therefore, it is hot topic to study for different families of graphs as well. Convex polytopes play an important role both in various branches of mathematics and in applied areas, most notably in linear programming. In this paper, we determine the metric-based resolving parameter of line graph of a convex polytope Sn, and conclude that it has constant metric dimension but vary with the parity of n . This article presents a measurement of the line graph of a convex polytope, denoted as ( S n ) . The subsequent section provides the metric dimension of the resulting graph. There are two scenarios pertaining to the metric dimension of a selected graph with respect to the metric dimension. The metric dimension of even cycle-based convex polytopes is three, whereas for other values, the metric dimension is four.

A bidirectional interpretable compound-protein interaction prediction framework based on cross attention

The identification of compound-protein interactions (CPIs) plays a vital role in drug discovery. However, the huge cost and labor-intensive nature in vitro and vivo experiments make it urgent for researchers to develop novel CPI prediction methods. Despite emerging deep learning methods have achieved promising performance in CPI prediction, they also face ongoing challenges: (i) providing bidirectional interpretability from both the chemical and biological perspective for the prediction results; (ii) comprehensively evaluating model generalization performance; (iii) demonstrating the practical applicability of these models. To overcome the challenges posed by current deep learning methods, we propose a cross multi-head attention oriented bidirectional interpretable CPI prediction model (CmhAttCPI). First, CmhAttCPI takes molecular graphs and protein sequences as inputs, utilizing the GCW module to learn atom features and the CNN module to learn residue features, respectively. Second, the model applies cross multi-head attention module to compute attention weights for atoms and residues. Finally, CmhAttCPI employs a fully connected neural network to predict scores for CPIs. We evaluated the performance of CmhAttCPI on balanced datasets and imbalanced datasets. The results consistently show that CmhAttCPI outperforms multiple state-of-the-art methods. We constructed three scenarios based on compound and protein clustering and comprehensively evaluated the model generalization ability within these scenarios. The results demonstrate that the generalization ability of CmhAttCPI surpasses that of other models. Besides, the visualizations of attention weights reveal that CmhAttCPI provides chemical and biological interpretation for CPI prediction. Moreover, case studies confirm the practical applicability of CmhAttCPI in discovering anticancer candidates.

AutoDDI: Drug-Drug Interaction Prediction With Automated Graph Neural Network.

Drug-drug interaction (DDI) has attracted widespread attention because when incompatible drugs are taken together, DDI will lead to adverse effects on the body, such as drug poisoning or reduced drug efficacy. The adverse effects of DDI are closely determined by the molecular structures of the drugs involved. To represent drug data effectively, researchers usually treat the molecular structure of drugs as a molecule graph. Then, previous studies can use the handcrafted graph neural network (GNN) model to learn the molecular graph representations of drugs for DDI prediction. However, in the field of bioinformatics, manually designing GNN architectures for specific molecular structure datasets is time-consuming and depends on expert experience. To address this problem, we propose an automatic drug-drug interaction prediction method named AutoDDI that can efficiently and automatically design the GNN architecture for drug-drug interaction prediction without manual intervention. To this end, we first design an effective search space for drug-drug interaction prediction by revisiting various handcrafted GNN architectures. Then, to efficiently and automatically design the optimal GNN architecture for each drug dataset from the search space, a reinforcement learning search algorithm is adopted. The experiment results show that AutoDDI can achieve the best performance on two real-world datasets. Moreover, the visual interpretation results of the case study show that AutoDDI can effectively capture drug substructure for drug-drug interaction prediction.

Study for Some Eccentricity-based Topological Indices of Second Type of Dominating David-derived Network.

Dominating David-derived networks are widely studied due to their fractal nature, with applications in topology, chemistry, and computer sciences. The use of molecular structure descriptors is a standard procedure that is used to correlate the biological activity of molecules with their chemical structures, which can be useful in the field of pharmacology. This article's goal is to develop analytically closed computing formulas for eccentricitybased descriptors of the second type of dominating David-derived network. Thermodynamic characteristics, physicochemical properties, and chemical and biological activities of chemical graphs are just a few of the many properties that may be determined using these computation formulas. Vertex sets were initially divided according to their degrees, eccentricities, and cardinalities of occurrence. The eccentricity-based indices are then computed using some combinatorics and these partitions. Total eccentricity, average eccentricity, and the Zagreb index are distance-based topological indices utilized in this study for the second type of dominating David-derived network, denoted as D2(m). These calculations will assist the readers in estimating the fractal and difficult-tohandle thermodynamic and physicochemical aspects of chemical structure. Apart from configuration and impact resistance, the D2(m) design has been used for fundamental reasons in a variety of technical and scientific advancements.

Further Results for Some Eccentric Based Indices.

Topological indices have a significant role in determining a molecule's topology in the light of theoretical chemistry. This study aimed to find new bounds for some eccentric-based topological indices. We used comparison theorem for integrals. We found new bounds for the eccentric-based harmonic index and atom-bond connectivity index. Our results are better than previous results in the literature. This study has contributed to the chemical graph theory literature by giving a new method to find eccentric-based topological indices with lower and upper bounds.

Exploring physico-chemical properties of HIV/AIDS drugs using neighborhood topological indices of molecular graphs

In this study, we investigate the efficacy of neighborhood-degree-based topological indices in the modeling of drug properties pertinent to HIV/AIDS. By representing molecular structures as graphs, we delve deep into atom-level environments, uncovering intricate relationships between local topological attributes and theoretical characteristics. Through meticulous quantitative structure–property relationship analysis, we establish robust correlations between these indices and drug properties. This breakthrough augurs predictive insights in the realm of pharmaceutical research, reducing the need for exhaustive experimentation. Our research underscores the pivotal role played by neighborhood-degree-based topological indices in advancing drug discovery, offering a powerful tool that resonates with chemists and industry professionals. It marks a transformative step in the trajectory of pharmaceutical development, promising to redefine and enhance the future of drug design and innovation.