Let $x_1,\cdots x_{k+1}$ be real numbers, where $k\geq 1$. Prove that $k.\sum_{i=2}^{k+1}(x_1-x_i)^2 \geq \sum_{2\leq i,j\leq k+1, i<j}(x_i-x_j)^2$.
I am trying to apply Cauchy–Schwarz inequality. The inequality can be explained as follows : Let $S$ be star graph on vertices $\{1,2,\cdots,k+1\}$ with $1$ as the center. Let $K$ be the complete graph on vertices $\{2,3,\cdots,k+1\}$. The above inequality would then imply that $k.L_{S} - L_{K}$ is positive semi-definite. Here $L_G$ is the Laplacian associated with the graph $G$. This lemma is from this paper. [Lemma 2.10]
We need to prove that $$k^2x_1^2-2k\sum_{i=2}^{k+1}x_i+k\sum_{i=2}^{k+1}-(k-1)\sum_{i=2}^{k+1}+2\sum_{2\leq i<j\leq k+1}x_ix_j\geq0,$$
which is just $$\left(kx_1-\sum_{i=2}^{k+1}x_i\right)^2\geq0.$$ Done!