Why is it true $||Aj|| \le ||j||$? Where $A$ is $n \times n$ matrix and $j$ is a vector.
Edit: matrix $A$ is given by formula $a_{ij}=\frac{1}{\sqrt{d(i)\cdot d(j)}}$ if $ij \in E(G)$ and $0$ otherwise.
$A$ is called the degree adjacency matrix of graph $G$ and $d(i)$ is degree of vertex $i$.
It is certainly false. You can replace $A$ by $kA$ and let $k \to \infty$ to get a contradiction.