Sharp Poincare-Wirtinger inequalities on complete graphs

Let $K_n=(V,E)$ be the complete graph with $n\ge 3$ vertices (here $V$ and $E$ denote the set of vertices and edges of $K_n$ respectively). We find the optimal value ${\mathbf{C}}_{n,p}$ such that the inequality $$\mathrm{\parallel }f-m_f{\mathrm{\parallel }}_p\le {\mathbf{C}}_{n,p}{\mathrm{V}\mathrm{a}\mathrm{r}}_{p}f$$ holds for every $f:V\to \mathbb{R},$ where ${\mathrm{V}\mathrm{a}\mathrm{r}}_p$ stands for the $p$-variation, and $m_f$ stands for the average value of $f$, for all $p\in [1,3+{\delta }_n^1)\cup (3+{\delta }_n^2,+\mathrm{\infty })$, for ${\delta }_n^1=\frac{1}{2n^2\log (n)}+O(1\mathrm{/}{n}^3)$ and ${\delta }_n^2=\frac{2}{n}+O(1\mathrm{/}{n}^2)\mathrm{.}$ Moreover, we characterize all the maximizer functions in that case. The behavior of the maximizers is different in each of the intervals $(1,2)$, $(2,3+{\delta }_n^1)$ and $(3+{\delta }_n^2,\mathrm{\infty })\mathrm{.}$