How can I show that the spectral radius of a graph $G$ is less than or equal to its maximum degree?
I have an unweighted graph $G=(V,E)$, where $V$ is its vertex set and $E$ its edge set. The degree of a vertex $x$ is $d_x$ and $D(G)$ if defined as $D(G)=\max\{d_x,x\in V\}$. In the paper I'm reading, the spectral radius is defined as $R(G)=\limsup_{n \to \infty}(A^n(x,y))^{1/n}$, where $A^n$ is the $n^{\text{th}}$ power of the adjacency matrix of $G$, but usually it's defined as the largest of its eigenvalues in absolute value. I want to see the intuition behind the inequality $R(G) \le D(G)$.