Let G be a simple molecular graph without directed and multiple edges and without loops. The vertex and edge-sets of G are denoted by V(G) and E(G), respectively. Suppose G is also a connected molecular graph and let u, v is an element of V(G) be two vertices. The harmonic index H(G) of G is defined as the sum of the weights 2(d(u)+d(v))(-1) of all edges in E(G), where d(v) is the degree of a vertex v in G which is defined as the number of vertices of G adjacent to v. The harmonic polynomial of G is defined as H(G, x) = Sigma(e=uv is an element of E(G)) 2x((du+dv-1)) and there is the following nice relation between these two notions H(G) = integral(1)(0) H(G, x)dx. In this paper, we present an explicit formula for the harmonic indices and harmonic polynomials of carbon nanocones CNCk[n].

The resistance distance is an intrinsic metric on graphs that have been extensively studied by many physicists and mathematicians. The resistance distance between two vertices of a simple connected graph $G$ is equal to the resistance between two equivalent points on an electrical network, constructed to correspond to $G$ , with each edge being replaced by a unit resistor. Hypercube $Q_{n}$ is one of the most efficient and versatile topological structures of the interconnection networks, which received much attention over the past few years. The folded $n$ -cube graph is obtained from hypercube $Q_{n}$ by merging vertices of the hypercube $Q_{n}$ that are antipodal, i.e., lie at a distance $n$ . Folded $n$ -cube graphs have been studied in parallel computing as a potential network topology. The folded $n$ -cube has the same number of vertices but half the diameter as compared to hypercubes which play an important role in analyzing the efficiency of interconnection networks. We intend is to minimize the diameter. In this study, we will compute the resistance distance between any two vertices of the folded $n$ -cube by using the symmetry method and classic Kirchhoff’s equations. This method is beneficial for distance-transitive graphs. As an application, we will also give an example and compute the resistance distance in the Biggs-Smith graph, which shows the competency of the proposed method.

Neurodegenerative diseases (NDDs) have received considerable interest from scientists for a long time due to their multifactorial nature. Alzheimer’s disease (AD) is of particular importance among pathologies, and despite approved drugs for its treatment, there is no effective pharmacological therapy to stop, halt, or repair neurodegeneration. The U.S. Food and Drug Administration (FDA) has approved certain medications to treat AD, including galantamine, donepezil, rivastigmine, memantine, tacrine, suvorexant, brexpiprazole, butein, and Licochalcone A. Topological indices and quantitative structure‐property relationships (QSPRs) are indispensable in drug discovery. They allow researchers to analyze, compare, and predict the properties of chemical compounds, thereby expediting the identification of promising drug candidates while minimizing experimental costs and efforts. Regression models are vital in QSPR analysis, especially when dealing with topological indices. They facilitate quantifying the relationship between chemical structure and properties, thereby facilitating drug design, material discovery, and other chemistry‐related applications. The objective of this study is to examine the efficacy of descriptors in correlating the physicochemical features of molecules associated with Alzheimer’s disease. We will use the computational method to calculate degree‐related, distance‐related, and eccentricity‐based topological indices for any chemical graph. QSPR models are developed utilizing degree‐based, distance‐based, and eccentricity‐based topological indices to estimate some physicochemical properties of AD drugs, including molecular weight (MW), boiling point (BP), topological polar surface area (TPSA), complexity (C), polarizability (P), and refractive index (R). The QSPRS studies are obtained using the linear regression technique. The present study found that the topological indices Randic index R, first Zagreb index M₁, and atom‐bond connectivity index ABC provide valuable insights into the structure‐activity relationships of different drugs and help in designing more effective combinations for treating Alzheimer’s disease.