Some remarks on the general zeroth--order Randi\'c coindex

Abstract

Let $G =(V,E)$, $V=\{v_1,v_2,\ldots, v_n\}$,  be a simple connected  graph of order $n$ and size $m$, without isolated vertices. Denote by  $d_1 \ge d_2 \ge \cdots \ge d_n$, $d_i=d(v_i)$  a sequence of vertex degrees of $G$.  The general zeroth--order Randi\'c index is defined as $^0\!R_{\alpha}(G)=\sum_{i=1}^nd_i^{\alpha}$, where $\alpha$ is an arbitrary real number. The corresponding general zeroth--order Randi\'c coindex is defined via $^0\!\bar R_{\alpha}(G)=\sum_{i=1}^n (n-1-d_i)d_i^{\alpha}$.  Some new bounds for the general zeroth--order Randi\'c coindex and  relationship between $^0\!\bar R_{\alpha}(\bar G)$ and $^0\!\bar R_{\alpha-1}(\bar G)$ are obtained. For a particular values of parameter $\alpha$ a number of new bounds for different topological coindices are obtained as corollaries.