Let's consider $A$ to be a square symmetric matrix whose entries are non-negative real numbers that sums to one. Even more, we shall consider its diagonal elements to be equal to zero. Under which conditions $A$ is non-negative-definite?
For the sake of good understanding, I attached an example:
$$A = \left[ \begin{array}{ccc} 0 & 0.1 & 0.25 \\ 0.1 & 0 & 0.15 \\ 0.25 & 0.15 & 0 \end{array} \right]$$
I thank you in advance for any contribution.
The only circumstance under which such a matrix can be non-negative definite is if it is the zero matrix.
In particular, suppose that $i,j$ are such that $a_{ij} \neq 0$. Then $A$ has the principal submatrix $$ A[\{i,j\}] = \pmatrix{0&a_{ij}\\a_{ij} & 0} $$ Then $\det A[\{i,j\}] = -a_{ij}^2 < 0$, which means that this principal submatrix fails to be non-negative definite. So, $A$ must fail to be non-negative definite.