The rapid growth of graph theory has sparked interest among analysts, driven by its diverse applications in mathematical chemistry. Closed-form solutions enable rapid property prediction without expensive simulations. This study delves into the second type of dominating David-derived network, which play a vital role in pharmaceutical development, hardware engineering, and system administration. We examine the topological features of the network, calculating distance-based indices like eccentricity measures and the eccentricity based Zagreb indices. Our findings offer novel perspectives on the structural attributes of dominating David-derived network, highlighting their potential impact across various disciplines.

Chemical graph theory establishes a connection between the properties of molecules and their corresponding molecular graphs. A topological index is a graph invariant that characterizes the graph’s structure and remains unaffected by graph automorphisms. In chemical graph theory, degree-based topological indices are particularly significant, offering crucial insights into the structural features of molecules. In this work, we introduce the maximum independent bond set polynomial MIBSP(H; x, y), a powerful tool for deriving various degree-based topological indices. We specifically apply MIBSP(H; x, y), to the chemical graphs of several painkiller molecules, including Aspirin, Paracetamol, Caffeine, Ibuprofen, Phenacetin, and Salicylic acid. The degree-based topological indices derived from these polynomials provide a deeper understanding of the molecular structures and their potential applications in pharmaceutical research.

Iterative graph have several applications in social network analysis, optimization problems, machine learning, and game theory. Such graphs are also commonly used in chemistry, physics, and mathematics. In this article, we derive the M-polynomial for the fractal growth patterns of benzene (F GBn, n ≥ 1), the Pythagoras tree (P Tn, n ≥ 1), and the benzene dendrimer (DBn, n ≥ 2). Moreover, we compute some degree-based topological indices based on the Mpolynomials, such as the first Zagreb index, the second Zagreb index, the modified second Zagreb index, the general Randi´c index, the harmonic index, the inverse sum index, and the symmetric division degree index. Finally, we presented our work graphically and compared the sketches of M-polynomials and degree-based topological indices.

A chemical network is numerically represented by a topological index in chemical graph theory. As opposed to its chemical representation, a topological descriptor correlates with specific physical properties of the underlying chemical molecules. In this article, third type of hex-derived networks HDN3(r), THDN3(r), are described. The goal of this study is to develop some updated and closed formulas based on multiplicative graph invariants. Such as ordinary geometric-arithmetic (OGA), general version of harmonic index (GHI), sum connectivity index (SI), general sum connectivity index (GSI), first and second Gourava and hyper-Gourava indices, Shegehalli and Kanabur indices, first generalized version of Zagreb index (GZI), and forgotten index (FI) for the hexderived HDN3(r), THDN3(r), networks. Moreover, various types of edge for computing have been discovered and analyzed along with the order and size. The calculation of multiplicative topological features in networks that are generated from hexagonal structures is the main task of this work. Gaining more insight into the structural characteristics and possible uses of these networks requires examining the interaction between topological aspects and multiplication processes. To interpret the chemical compounds’, physical and biological attributes, we can integrate the analysis of the networks stated above with the chemical compounds and their graphical structures. These results can be utilized to evaluate the biological and physio-chemical activities of compounds.

A topological index is a numerical representation of a chemical network in chemical graph theory. Similar to their chemical representation, a topological descriptor identifies specific the physical attributes of the underlying chemical compounds.We describe the third type of hex-derived networks rectangular hex-derived network RHDN3, chain hex-derived network CHDN3, in our work (r). In this paper, modified and efficient formulas based on multiplicative graph invariants will be constructed. Such as ordinary geometric-arithmetic (OGA), general version of harmonic index (GHI), sum connectivity indes (SI), general sum connectivity index (GSI), first and second Gourava and hyper- Gourava indices, Shegehalli and Kanabur indices, first generalised version of Zagreb index (GZI) and forgotten index (FI) for the subdivided hex -derived RHDN3(r), CHDN3(r) networks. In order to develop some new formulas we the compute the multiplicative topological properties. This study found several types of edges for computing and also discussed the order and size.To properly appreciate the chemical compounds, physical and biological characteristics we can combine the study of the networks described above with the chemical compounds and their graphical structures. These observations have the potential to evaluate bio and physiological activities.

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.

Chemical descriptors are numeric numbers that contain a basic chemical structure and describe the structure of a graph. A graph’s topological indices are linked to its chemical characteristics. Biological activity of chemical compounds can be predicted using topological indices. Numerous chemical indices have been developed in theoretical chemistry, including the Zagreb index, the Randić index, the Wiener index, and many others. In this paper, we compute the exact results for the Randić, Zagreb, Harmonic, augmented Zagreb, atom-bond connectivity, and geometric-arithmetic indices for the Benzenoid networks theoretically.

Topological indices are empirical features of graphs that characterize the topology of the graph and, for the most part, are graph independent. An important branch of graph theory is chemical graph theory. In chemical graph theory, the atoms corresponds vertices and edges corresponds covalent bonds. A topological index is a numeric number that represents the topology of underline structure. In this article, we examined the topological properties of prism octahedron network of dimension m and computed the total eccentricity, average eccentricity, Zagreb eccentricity, geometric arithmetic eccentricity, and atom bond connectivity eccentricity indices, which are used to utilize the distance between the vertices of a prism octahedron network.

A branch of graph theory that makes use of a molecular graph is called chemical graph theory. Chemical graph theory is used to depict a chemical molecule. A graph is connected if there is an edge between every pair of vertices. A topological index is a numerical value related to the chemical structure that claims to show a relationship between chemical structure and various physicochemical attributes, chemical reactivity, or, you could say, biological activity. In this article, we examined the topological properties of a planar octahedron network of m dimensions and computed the total eccentricity, average eccentricity, Zagreb eccentricity, geometric arithmetic eccentricity, and atom bond connectivity eccentricity indices, which are used to determine the distance between the vertices of a planar octahedron network.

Cancer is a disease that causes abnormal cell formation and spreads throughout the body, causing harm to other organs. Breast cancer is the most common kind among many of cancers worldwide. Breast cancer affects women due to hormonal changes or genetic mutations in DNA. Breast cancer is one of the primary causes of cancer worldwide and the second biggest cause of cancer-related deaths in women. Metastasis development is primarily linked to mortality. Therefore, it is crucial for public health that the mechanisms involved in metastasis formation are identified. Pollution and the chemical environment are among the risk factors that are being indicated as impacting the signaling pathways involved in the construction and growth of metastatic tumor cells. Due to the high risk of mortality of breast cancer, breast cancer is potentially fatal, more research is required to tackle the deadliest disease. We considered different drug structures as chemical graphs in this research and computed the partition dimension. This can help to understand the chemical structure of various cancer drugs and develop formulation more efficiently.