Sharp bounds for the general Randić index of transformation graphs

2020 ◽  
Vol 39 (5) ◽  
pp. 7787-7794
Author(s):  
Muhammad Imran ◽  
Shehnaz Akhter ◽  
Hani Shaker

Inequalities are a useful method to investigate and compare topological indices of graphs relatively. A large collection of graph associated numerical descriptors have been used to examine the whole structure of networks. In these analysis, degree related topological indices have a significant position in theoretical chemistry and nanotechnology. Thus, the computation of degree related indices is one of the successful topic of research. Given a molecular graph H , the general Randić connectivity index is interpreted as R α ( H ) = ∑ ℛ ∈ E ( H ) ( deg H ( a ) deg H ( b ) ) α , with α is a real quantity. Also a graph transformation of H provides a comparatively simpler isomorphic structure with an ease to work on different chemical properties. In this article, we determine the sharp bounds of general Randić index of numerous graph transformations, such that semi-total-point, semi-total-line, total and eight individual transformations H fgh , where f, g, h ∈ {+ , -} of graphs by using combinatorial inequalities.

2020 ◽  
Vol 2020 ◽  
pp. 1-17
Author(s):  
Xu Li ◽  
Maqsood Ahmad ◽  
Muhammad Javaid ◽  
Muhammad Saeed ◽  
Jia-Bao Liu

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.


2021 ◽  
Vol 6 (12) ◽  
pp. 13887-13906
Author(s):  
Fei Yu ◽  
◽  
Hifza Iqbal ◽  
Saira Munir ◽  
Jia Bao Liu ◽  
...  

<abstract><p>In the chemical industry, topological indices play an important role in defining the properties of chemical compounds. They are numerical parameters and structure invariant. It is a proven fact by scientists that topological properties are influential tools for interconnection networks. In this paper, we will use stellation, medial and bounded dual operations to build transformed networks from zigzag and triangular benzenoid structures. Using M-polynomial, we compute the first and second Zagreb indices, second modified Zagreb indices, symmetric division index, general Randic index, reciprocal general Randic index. We also calculate atomic bond connectivity index, geometric arithmetic index, harmonic index, first and second Gourava indices, first and second hyper Gourava indices.</p></abstract>


2007 ◽  
Vol 155 (8) ◽  
pp. 1044-1054 ◽  
Author(s):  
Yumei Hu ◽  
Xueliang Li ◽  
Yongtang Shi ◽  
Tianyi Xu

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