>[!success] Theorem >If $\mathbf{A}$ is a [[Matrices|non-negative matrix]] (e.g. [[Adjacency Matrix]] with all positive weights) then: >- $\lambda_1$ is a real [[Eigenvalue Problem|eigenvalue]] with multiplicity 1 if the graph is connected. > - Corresponding eigenvector has all-positive entries. >- For any other eigenvalue $\lambda_{1}\geq |\lambda_k|$. >- If [[Graph|graph]] is also undirected ($\mathbf{A}$ symmetric), all eigenvalues are real and > - $\lambda_{1}\geq \lambda_{2}\geq ... \geq \lambda_{N}$ > - $\lambda_{1}\geq \lambda_{2}$ if graph is connected >- Multiplicity of first eigenvalue corresponds to number of connected components.