Eigenvector centrality for network analysis

Wikipedia gives a succinct explanation for the eigenvector centrality measure. Basically the centrality score of vertex $v$ is

where $a_{v,t}$ is an element of the adjacency matrix $A$ with a value of 1 for connection between $v$ and $t$, and 0 for no direct connection between $v$ and $t$.  We can move $\lambda$ to the left side of the above equation, and substitutie elements with vector and matrix,

Hence, $\lambda$ is the eigen value of matrix $A$. 

One of my nagging question is why it is sufficient to focus only on the greatest eigen value. According to Wikipedia entry, because all the elements in the eigen vector ought to be positive, the Perron-Frobenius theorem indicates that only the greatest eigen value can satisfy this requirement. Elements in the corresponding dominant eigen vector (I am not sure of this term) give the centrality measure for each row (vertice). 

Wikipedia mentions that matrix $A$ can be generalized to weighted matrix. 

Reference: 

http://en.wikipedia.org/wiki/Eigenvector_centrality#Eigenvector_centrality

Perron–Frobenius theorem