Abstract
A topological invariant is a numerical parameter associated with molecular graph and plays an imperative role in the study and analysis of quantitative structure activity/property relationships (QSAR/QSPR). The correlation between the entire π-electron energy and the structure of a molecular graph was explored and understood by the first Zagreb index. Recently, Liu et al. (2019) calculated the first general Zagreb index of the F-sum graphs. In the same paper, they also proposed the open problem to compute the general Randić index RαΓ=∑uv∈EΓdΓu×dΓvα of the F-sum graphs, where α∈R and dΓu denote the valency of the vertex u in the molecular graph Γ. Aim of this paper is to compute the lower and upper bounds of the general Randić index for the F-sum graphs when α∈N. We present numerous examples to support and check the reliability as well as validity of our bounds. Furthermore, the results acquired are the generalization of the results offered by Deng et al. (2016), who studied the general Randić index for exactly α=1.
Highlights
Suppose the ordered pair (V(Γ), E(Γ)) denotes a finite, simple, and connected molecular graph Γ. e set represented by V(Γ) is the vertex set and the set denoted by E(Γ), disjoint from V(Γ), is the edge set
Two primary parameters known as the order and the size of graph Γ are denoted by n and e
Randicindex of F-sum graphs for α ∈ R and following results are intimately tied with generalized binomial and trinomial theorems
Summary
Suppose the ordered pair (V(Γ), E(Γ)) denotes a finite, simple, and connected molecular graph Γ. e set represented by V(Γ) is the vertex set and the set denoted by E(Γ), disjoint from V(Γ), is the edge set. Suppose the ordered pair (V(Γ), E(Γ)) denotes a finite, simple, and connected molecular graph Γ. Two primary parameters known as the order (total number of vertices) and the size (total number of edges) of graph Γ are denoted by n and e. A cycle Cm is a simple graph with same order and size m in such a way that each vertex has degree 2. To probe and study the chemical, structural, and physical properties of the molecular graphs within the subject of chemical graph theory, several TIs are proposed and intensely investigated. Ese TIs helped to study the chemical reactivities and physical features such as heat of evaporation and formation, boiling, melting and freezing point, volume of air and vapor pressure, surface tension and density, and critical temperature of the chemical compounds that are involved in the molecular graphs. For further reading regarding development and applications of TIs, the readers are referred to [1,2,3,4,5,6,7,8,9,10]
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