In order to further study the resonance theory of plane bipartite graph and its related properties, firstly, some theorems and recurrence relations are obtained by studying the Clar covering polynomials of plane bipartite graph. Furthermore, the relationship between certain parameters in the Clar covering polynomials and topological indices in chemistry are obtained, and explicit expression for catacondensed plane bipartite graph is derived. The Clar covering polynomial of molecular graphs is a method for describing the electronic structure in conjugated systems. By studying the Clar covering polynomials of plane bipartite graphs, it is possible to understand the electronic structure of chemical molecules, predict their chemical properties and reaction behavior, and design new molecular structures.

M polynomials and NM polynomials are integral concepts in polynomial graph theory. M polynomials, like the matching polynomial, provide insights into matching structures in graphs, while NM polynomials extend this to non-matching edges. These tools are crucial in understanding graph properties and are applied in diverse fields such as network design and chemistry. Often topological indices are derived from these polynomials, which are used in Quantitative Structure Activity Relations (QSAR)/ Quantitative Structure Property Relations (QSPR) studies that have applications in protein structure analysis, network communication optimization, drug design, drug discovery, pharmacokinetics, etc. In this paper, we have defined M, NM polynomials for reverse, reduced reverse, neighborhood reverse and neighborhood reduced reverse topological indices. And we have derived closed form expressions for topological indices from these polynomials for Y junction nanotubes. Also as an application we have developed a QSPR model for an important thermodynamic property, viz., bond energy, of Y-junction nanotubes using indices derived from the defined polynomials Mr, Mrr, NMr, NMrr. The QSPR model developed here is statistically robust with an R2 of 0.999, with marginal error and high F value, hence serves as a justification to our new definitions for more polynomials, topological índices to the ever increasing set.

In this paper we determine the recently introduced omega index of some Sierpin ́ski-like graphs such as Sierpin'ski graph, Extended Sierpin ́ski graphs such as S(n,k) and extended version. Further, we obtain a relation between several topological indices with omega index for these Sierpinski-like graphs. The degree sequence of a graph is the list of its vertex degrees arranged in usually increasing order. Many properties of the graphs realized from a degree sequence can be deduced by means of recently introduced graph invariant called omega invariant. We used the definitions of some well-known Sierpin ́ski-like graphs such as S(n,k), and extended Sierpin ́ski graphs together with the list of degree sequences of these graphs. Naturally, the vertex degree and edge degree partitions are used. As the main theme of the paper is the omega invariant, we frequently used the definition and fundamental properties of this very new invariant for our calculations. Several well-known combinatorial and graph theoretical proof techniques are used in deriving the proofs of results given here. In this paper, we have determined the general formula for the omega index of Sierpin ́ski graphs in terms of k and n, general formula for the omega index of extended Sierpin ́ski graphs such as and . The general formula for Zagreb indices, forgotten index, reverse Zagreb indices and reverse forgotten index of extended Sierpin ́ski graphs are obtained. Also, we obtain various relations between several topological indices and omega invariant for Sierpin ́ski-like graphs. Thus, establishing some new connections between graph theory, fractals and chemical graph theory. We have determined the general formula for omega indices of some Sierpin ́ski-like graphs such as Sierpin ́ski graph S(n,k), extended Sierpin ́ski graphs such as S^+ (n,k)and S^(++) (n,k). Further, we have obtained various relations between several important topological indices such as Zagreb indices, forgotten index, reverse Zagreb indices and reverse forgotten index with omega index for these Sierpin ́ski-like graphs.

Drug target interactions (DTIs) prediction is a crucial task in contemporary drug development. However, traditional drug target prediction methods have problems such as long-term consumption and low accuracy. In this study, we present an advanced drug–target prediction model that integrates enhanced multi-granularity scanning with a deep forest architecture. First, by utilizing the SMOTE algorithm for data balancing, it ensures a more equitable representation of positive samples in the dataset. Second, a dynamic adaptive scanning method is introduced to effectively handle feature extraction across varying data scales. Third, a model, DMS-DF, incorporates XGBoost and CatBoost as base classifiers for improved performance. Our approach demonstrates superior performance in drug–target prediction, paving the way for more efficient drug discovery.

Graphical indices and quantitative structure-property relationships (QSPRs) are essential in drug discovery. Researchers utilize them to examine, contrast, and forecast the characteristics of chemical compounds, therefore accelerating the discovery of potential therapeutic candidates while reducing the need for extensive experimentation and expenses. Regression models are crucial in quantitative structure-property relationship (QSPR) analysis, particularly when working with graphical indices. They enable the measurement of the correlation between chemical structure and characteristics, aiding in medication design, material discovery, and other chemistry-related applications. In this study, we employ newly approved medications by the United States Food and Drug Administration (FDA) in 2022 as test compounds. Several of these items contain active substances that have yet to be previously approved by the FDA, either in their pure form or as combined components. Patients suffering from various untreated illnesses frequently experience substantial therapeutic benefits from the notable innovative contributions of these items. This study focuses on developing quantitative structure-property relationship (QSPR) models. These models utilize degree-based graphical indices to estimate certain physicochemical characteristics of newly FDA-approved pharmaceuticals. The properties of interest include molecular weight (MW), polarizability (P), molar refractivity (R) and density (D). The estimation is done using the linear regression approach.

A topological index is a numerical representation of a graph’s structure; it gives some important information related to a graph. Topological indices, like mathematical modeling of chemical compounds’ toxicological, pharmacologic, physio-chemical, and other features, are widely used in chemical investigation. Various topological indices have been defined in the literature based on degree, distance, spectrum, etc. In this paper, we consider several well-known degree-based indices: the sum-connectivity index, the atom-bond sum-connectivity index, the sombor index, the first Zagreb index, the forgotten topological index, and the first hyper-Zagreb index. We arrange the indices sequentially, with the help of the differences among them. Furthermore, we explore the association between the indices of the molecular graphs of octane isomers C8H18. We also establish the correlation between indices and some physical properties: entropy, total surface area, and acentric factor of octane isomers.