6. Let $n$ be a fixed integer, $n \geqslant 2$, determine the smallest constant $c$, such that the inequality $\sum_{1 \leqslant i \leqslant j \leqslant n} x_{i} x_{j}\left(x_{i}^{2}+x_{j}^{2}\right) \leqslant$ $c \cdot\left(\sum_{i=1}^{n} x_{i}\right)^{4}$ holds for all non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$, and determine the necessary and sufficient conditions for equality.
6. Obviously, the inequality always holds when $x_{1}=x_{2}=\cdots=x_{n}=0$. If $x_{i}(i=1,2, \cdots, n)$ are not all 0, by homogeneity, we can assume $x_{1}+x_{2}+\cdots+x_{n}=1$. When $n=2$, $x_{1}+x_{2}=1$, the left side $=x_{1} x_{2}\left(x_{1}^{2}+x_{2}^{2}\right)=x_{1} x_{2}\left(1-2 x_{1} x_{2}\right)=$ $\frac{1}{2} \cdot 2 x_{1} x_{2}\left(1-2 x_{1} x_{2}\right) \leqslant \frac{1}{8}$, equality holds if and only if $x_{1}=x_{2}=\frac{1}{2}$.