On the bounds of zeroth--order general Randic index

Abstract

The zeroth--order general Randi\' c index, $^0\!R_{\alpha}(G)$, of a connected graph $G$, is defined as $^0\!R_{\alpha}(G)=\sum_{i=1}^{n}d_{i}^{\alpha}$,
where $d_{i}$ is the degree of the vertex $v_{i}$ of $G$ and $\alpha$ arbitrary real number. We consider linear combinations of the
$^0\!R_{\alpha}(G)$ of the form $ ^0\!R_{\alpha}(G) - (\Delta +\delta) ^0\!R_{\alpha -1}(G)+ \Delta\delta\, ^0\!R_{\alpha-2}(G)$ and
$^0\!R_{\alpha}(G)- 2a\,\, ^0\!R_{\alpha-1}(G)+ a^2\,\,{^0\!R_{\alpha-2}(G)}$, where $a$ is an arbitrary real number, and determine their bounds. As corollaries, various upper and lower bounds of $^0\!R_{\alpha}(G)$ and indices that represent some special cases of $^0\!R_{\alpha}(G)$ are obtained.