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The study centered on Quantitative Structure Property Relationship (QSPR) analysis with a focus on various graph energies, investigating drugs like Mefloquinone, Sertraline, Niclosamide, Tizoxanide, PHA-690509, Ribavirin, Emricasan, and Sofosbuvir. Employing computational modeling techniques, the research aimed to uncover the correlations between the chemical structures of these medications and their unique properties. The results illuminated the quantitative relationships between structural characteristics and pharmacological traits, advancing our predictive capabilities. This research significantly contributes to medication discovery and design by providing essential insights into the structure-property connections of these medicinal compounds. Notably, certain spectrum-based descriptors, such as positive inertia energy, adjacency energy, arithmetic-geometric energy, first zegrab energy, and the harmonic index, exhibited strong correlation coefficients above 0.999. In contrast, well-known descriptors like the Extended adjacency, Laplacian and signless Laplacian spectral radii, and the first and second Zagreb Estrada indices showed weaker performance. The article emphasizes the application of graph energies and a linear regression model to predict pharmacological features effectively, enhancing the drug discovery process and aiding in targeted drug design by elucidating the relationship between molecular structure and pharmacological characteristics.

Let K be a 2-torsion free unital ring and D(K) be dihedron algebra over K. In the present article, we prove that every Lie triple derivation of D(K) can be written as the sum of the Lie triple derivation of K, Jordan triple derivation of K, and some inner derivation of D. We also prove that a generalized Lie triple derivation ϱ:D(K)→D(K) associated with the Lie triple derivation h:D(K)→D(K) exists if ϱ can be represented in the form ϱ(τ) = h(τ) + λτ, where λ lies in the center of D(K). We finally conclude that to obtain the complete algebra of the Lie triple derivation and generalized Lie triple of D(K), we first need to find the Lie triple derivation and Jordan triple derivation of K.

Graph energy is defined to be the p-norm of adjacency matrix associated to the graph for p = 1 elaborated as the sum of the absolute eigenvalues of adjacency matrix. The graph’s spectral radius represents the adjacency matrix’s largest absolute eigenvalue. Applications for graph energies and spectral radii can be found in both molecular computing and computer science. On similar lines, Inverse Sum Indeg, (ISI) energies, and (ISI) spectral radii can be constructed. This article’s main focus is the ISI energies, and ISI spectral radii of the generalized splitting and shadow graphs constructed on any regular graph. These graphs can be representation of many physical models like networks, molecules and macromolecules, chains or channels. We actually compute the relations about the ISI energies and ISI spectral radii of the newly created graphs to those of the original graph.

Around 1880, Lie introduced an idea of invariance of the partial differential equation (PDE) under one-parameter Lie group of transformation to find the invariant, similarity, or auto-model solutions. Lie symmetry analysis (LSA) provides us an algorithm to search for point symmetries for solving related linear systems for infinitesimal generators. Actually, point symmetries lead us to one-parameter family of solutions from a known solution. LSA is a program that provides us the exact solutions for the non-linear differential equations (DEs) in analogy of the program designed by Galois for algebraic polynomial equations. In this paper, we have carried out the LSA for computing the similarity solutions (symmetries) of the non-linear short pulse equation (SPE) for the cases when h(u) = eu, k(u) = uxx, h(u)=eun, and k(u) = uxx. In addition, an optimal system of one-dimensional sub-algebra has been set up. The reductions and invariant solutions for the generalized SPE are calculated corresponding to this optimal system as well. Reductions reduce the non-linear PDE or system of PDEs into non-linear reduced ordered ODE or system of PDEs. This helps to solve these systems of PDEs to reduced form. Graphical behavior of the transformed points of the 1-parameter solution functions have drawn.

The present article presents some new results relating to Atomic Bond Connectivity energies and Spectral radii of generalized splitting and generalized shadow graphs constructed on the basis of some fundamental families of cycle graph Cn, complete graph Kn and complete bipartite graph Kn,n referred as base graphs. In fact we relate the energies and Spectral radii of splitting and shadow graphs with the energies and Spectral radii of original graphs.

Scientific organizations are creating carbon nanotube-based composites like VC5C7[m,n] and HC5C7[m,n], which indicate significant response against voltage. Imbalance-based irregularity indices determine the degree of irregularity of a certain molecular structure and, as a result, determine the properties of a molecular substance. In this article, we aim to compute irregularity indices of two nanotubes, VC5C7[m,n] and HC5C7[m,n]. We produce formulas for the irregularity of these two nanotubes, which are functions depending on the parameters of the structure m and n. We compare our results graphically and conclude that VC5C7[m,n] is more irregular than HC5C7[m,n].