General Randic Index of Unicyclic Graphs and Its Applications to Drugs

In this work, we determine the maximum general Randic index (a general symmetric function of vertex degrees) for eta(0 )<= eta < 0 among all n-vertex unicyclic graphs with a fixed maximum degree Delta and the maximum and the second maximum general Randic index for eta(0 )<= eta < 0 among all n-vertex unicyclic graphs, where eta(0 )approximate to -0.21. We establish sharp inequalities and identify the graphs attaining the inequalities. Thereby, extremal graphs are obtained for the general Randic index, and certain open gaps in the theory of extremal unicyclic graphs are filled (some open problems are provided). We use computational software to calculate the Randic index for the chemical trees up to order 7 and use the statistical (linear regression) analysis to discuss the various applications of the Randic index with the physical properties of drugs on the said chemical trees. We show that the Randic index is better correlated with the heat of vaporization for these alkanes.