Why is it true $||Aj|| \le ||j||$? Where $A$ is $n \times n$ matrix and $j$ is a vector.
Matrix $A$ is called the degree adjacency matrix of graph $G$ and is given by formula $a_{ij}=\frac{1}{\sqrt{d(i)\cdot d(j)}}$ if $ij \in E(G)$ and $0$ otherwise.
And $d(i)$ is degree of vertex $i$.
Note that $v = (\sqrt{d(1)},\dots,\sqrt{d(n)})$ is a vector with positive entries satisfying $Av = v$. It follows by the Perron-Frobenius theorem that $1$ is the largest eigenvalue of $A$. Because $A$ is a symmetric matrix, it follows that the spectral norm of $A$ is $1$, which is to say that $\|Aj\| \leq \|j\|$ for all vectors $j$.