Degree sequence measurements on graphs have attracted a lot of research interest in recent decades. Multiplying the degrees of adjacent vertices in graph $\Omega$ provides the multiplicative first Zagreb index of a graph. In the context of graph theory, the generalized multiplicative first Zagreb index of a graph $\Omega$ is defined as the product of the sum of the $\alpha$th powers of the vertex degrees of $\Omega$, where $\alpha$ is a real number such that $\alpha \ne 0$ and $\alpha \ne 1$. The focus of this work is on the extremal graphs for several classes of graphs including trees, unicyclic, and bicyclic graphs, with respect to the generalized multiplicative first Zagreb index. In the initial step, we identify a set of operations that either increases or decreases the generalized multiplicative first Zagreb index for graphs. We then involve analysis of the generalized multiplicative first Zagreb index achieving sharp bounds by characterizing the maximum or minimum graphs for those classes. We present applications of the generalized multiplicative first Zagreb index ${\Pi }_1^{\alpha }$ for predicting the $\pi$-electronic energy $E_{\pi }\left(\beta \right)$ of benzenoid hydrocarbons. In particular, we answer the question concerning the value of $\alpha$ for which the predictive potential of ${\Pi }_1^{\alpha }$ with $E_{\pi }$ for lower benzenoid hydrocarbons is the strongest. In fact, our statistical analysis delivers that ${\Pi }_1^{\alpha }$ correlates with $E_{\pi }$ of lower benzenoid hydrocarbons with correlation coefficient $\rho =-0.998$, if $\alpha =-0.00496$. In QSPR modeling, the value $\rho =-0.998$ is considered to be considerably significant.

Helmholtz’s problem helps us to completely understand how sound behaves in a cylinder that is closed from one of its ends and opened at another. This paper aims to employ some novel convergence results to the Helmholtz problem with mixed boundary conditions and demonstrate the existence and uniqueness of the solution by applying graph-controlled contractions. For this purpose, we enunciate graphically Reich type and graphically Ćirić type contractions in the realm of graphical-controlled metric type spaces. In our study, we showcase the existence and uniqueness of fixed point results by employing these graphical contractions. This is demonstrated through extensive examples that a graphicalcontrolled metric-type space is distinct from traditional controlled metric-type spaces. We also exhibit an example of a graphically Reich contraction that is not a classical Reich contraction. Similarly, a decent example of graphical Ćirić contraction is presented, which is distinct from the classical Ćirić contraction. Concrete illustrative examples solidify our theoretical framework.

In this manuscript, we propose the notion of a strong extended s-suprametric space, a novel extension that outperforms both s-suprametric and extended suprametric spaces. It looks into the aspects of open and closed ball topologies within this structure. It also investigates the concepts of existence and uniqueness using basic contractions viz. Banach and Kannan contractions. Illustrative examples demonstrate how the strong extended s-suprametric space outperforms its extended equivalent. Our examples demonstrate the presence and distinctness of fixed points in this scenario. Furthermore, exploiting these newly launched results, the manuscript investigates the analysis of a boundary value problem, including diffusing chemical material constrained between parallel walls with related concentrations at the boundaries, taking into account supplied raw density and recognized absorbing coefficients. It also applies these insights to a nonlinear boundary value issue involving satellite web coupling in which a thin sheet joins two cylindrical spacecraft. This coupling causes nonlinearity, resulting in a separate boundary value issue influenced by radiation effects within the satellites.