Some inequalities for general zeroth--order Randi\'c index

Abstract

Let $G=(V,E)$, $V=\{v_1,v_2,\ldots, v_n\}$, be a simple connected
graph with $n$ vertices, $m$ edges and vertex degree sequence
$\Delta=d_1 \ge d_2 \ge \cdots \ge d_n=\delta>0$, $d_i=d(v_i)$.
General zeroth--order Randi\'c index of $G$ is defined as
$^{0}\!R_{\alpha}(G)=\sum_{i=1}^n d_i^{\alpha}$, where
$\alpha$ is an arbitrary real number. In this paper we establish
relationships between $^{0}\!R_{\alpha}(G)$ and
$^{0}\!R_{\alpha-1}(G)$ and obtain new bounds for
$^{0}\!R_{\alpha}(G)$. Also, we determine relationship between $^{0}\!R_{\alpha}(G)$,
$^{0}\!R_{\beta}(G)$ and $^{0}\!R_{2\alpha-\beta}(G)$, where $\alpha$ and
$\beta$ are arbitrary real numbers. By the appropriate choice of parameters $\alpha$ and $\beta$, a number of old/new inequalities for different
vertex--degree--based topological indices are obtained.