Extremal zeroth-order general Randić index of thorn graphs. (Q2918560)

The zeroth-order Randić index of \(G\) with vertex set \(V\) is defined as \(R_\alpha ^0(G) = \sum _{v \in V} d_v^{\alpha }\), where \(\alpha \) is an arbitrary real number and \(d_v\) denotes the degree of vertex \(v\). The thorn graph \(G^{*}\) of \(G\) is obtained by attaching \(d_G(v)\) new pendant edges to each vertex \(v\) in \(G\), and the thorn graph of a tree is called a thorn tree. In this paper the authors determine the zeroth-order general Randić index for a class of thorn trees and characterize the extremal trees in this class. They then express \(R_{\alpha }^0(G^{*})\) in terms of \(R_{\alpha }^0(G)\) for an arbitrary graph \(G\).