now establish results relating graphs and eigenvalues of matrices:

Let $A$ be a stochastic matrix of size $N$, if $$\forall i,j \in \{1,...,N\}, i\xrightarrow{A} j \text{ or } j \xrightarrow{A} i,$$ holds then $1$ is an eigenvalue of $A$, and its multiplicity is $1$.

We suppose that $1$ is not an eigenvalue of multiplicity $1$ of $A$. Since $A$ is a stochastic matrix one has $A \mathbf e = \mathbf e$ with $\mathbf e$ given by , so $1$ is an eigenvalue of $A$. In particular, its order of multiplicity is bigger than 1, and there exists $X\in \mathbb{R}^N\setminus{\mathbb{R}}\mathbf e$ such that $A X = X$. Let P be a permutation matrix such that the coordinates of the vector $PX$ are ranked from the lowest to the highest. Let $A' = P^{T}A P$ and $X' = P^{T}X$. Let $n_l$ and $n_h$ be respectively the number of coordinates equals to the lowest and to the highest coordinates values. We have $n_l \geq 1$, $n_h \geq 1$ and $n_l + n_h \leq N$.

#### First case : $n_l + n_h = N$

$$\begin{aligned}
    A X = X & \iff A'X' = X'. \\
    &\implies 
        \forall i \in \{1,...,N\},\sum_{j=1}^N a_{ij}' X_j' = X_i'.\\
    