Non-vanishing elements and complex group algebras

Let $G$ be a finite group, and let $\mathrm{I}\mathrm{r}\mathrm{r}(G)$ denote the set of irreducible complex characters of $G$. An element $x$ of $G$ is said to be vanishing, if for some $\chi$ in $\mathrm{I}\mathrm{r}\mathrm{r}(G)$, we have $\chi (x)=0$. Also the element $x$ is called rational if $x$ is conjugate to $x^i$ for every integer $i$ co-prime to the order of $x$. We define the weight of $G$ as $\omega (G):=({\sum }_{\chi \in \mathrm{I}\mathrm{r}\mathrm{r}(G)}\chi (1){)}^2\mathrm{/}\mathrm{\mid }G\mathrm{\mid }$. In this paper, we show that for every rational non-vanishing element $x\in G$, the order of $C_G(x)$ is at least $\omega (G)$.