Let $A = D + \begin{bmatrix} 0 & F \\ F^T & 0 \\ \end{bmatrix} $ be in the block form, where $D$ is diagonal and $F$ is $k \times (n-k)$. Let $A$ be doubly non-negative, i.e. both non-negative entry wise and positive semi definite. Then $D^{- \frac12}AD^{- \frac12}$ is also doubly non-negative.
$$D^{- \frac12}AD^{- \frac12} = \begin{bmatrix} I_k & C \\ C^T & I_{n-k} \\ \end{bmatrix}$$, where $C$ is $k \times (n-k)$.
Further more if $A$ is irreducible doubly non-negative, then to prove that $rank(A) \geq n-1$.
I am not getting clue how to proceed.