Extremal (n,n + 1)-graphs with respected to zeroth- order general Randić index

A (n, n + 1)-graph G is a connected simple graph with n vertices and n + 1 edges. If dv denotes the degree of the vertex v, then the zeroth-order general Randić index \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R_\alpha^0(G)$$\end{document} of the graph G is defined as \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sum\nolimits_{v\in V(G)}d_v^\alpha$$\end{document}, where α is a real number. We characterize, for any α, the (n,n + 1)-graphs with the smallest and greatest zeroth-order general Randić index.