9. Let $n(n \geqslant 2)$ be a fixed integer,
(1) Determine the smallest constant $c$ such that the inequality $\sum x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant c\left(\sum^{n} x_{i}\right)^{4}$ holds for all non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$;
(2) For this constant $c$, determine the necessary and sufficient conditions for equality to hold.
9. First find the range of values for $c$, then provide a proof, and finally point out the necessary and sufficient conditions for the equality to hold when $c$ takes its minimum value.

When $x_{1}=x_{2}=1, x_{3}=x_{4}=\cdots=x_{n}=0$,
$$
c \geqslant \frac{1}{16} \times 1 \times 1 \times\left(1^{2}+1^{2}\right)=\frac{1}{8} \text {. }
$$

Below, we prove that the inequality $\sum_{1 \leqslant i<j \in n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant \frac{1}{8}\left(\sum_{i=1}^{n} x_{i}\right)^{4}$ holds for all non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$. Note that $\left(\sum_{i=1}^{n} x_{i}\right)^{4}=\left(\sum_{k=1}^{n} x_{k}^{2}+2 \sum_{i \in i<j \in n} x_{i} x_{j}\right)^{2} \geqslant 4 \sum_{k=1}^{n} x_{k}^{2} \cdot 2 \sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}$ $=8 \sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j} \cdot \sum_{k=1}^{n} x_{k}^{2} \geqslant 8 \sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x^{2} i_{+} x_{j}^{2}\right)$. Therefore, $\sum_{1 \leqslant i<j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant \frac{1}{8}\left(\sum_{i=1}^{n} x_{i}\right)^{4}$.
Thus, the minimum value of $c$ is $\frac{1}{8}$. The equality holds if and only if two of the $x_{i}$ are equal, and the remaining $n-2$ $x_{i}$ are all 0.