Sharp Inequalities for maximal operators on finite graphs

Let $G=(V,E)$ be a finite graph and $M_G$ be the centered Hardy-Littlewood maximal operator defined there. We find the optimal value ${\mathbf{C}}_{\mathbf{G},\mathbf{p}}$ such that the inequality $${\text{Var}}_p(M_{G}f)\le {\mathbf{\text{C}}}_{G,p}{\text{Var}}_p(f)$$ holds for every $f:V\to \mathbb{R},$ where ${\text{Var}}_p$ stands for the $p$-variation, when: (i) $G=K_n$ (complete graph) and $p\in [\frac{\log (4)}{\log (6)},\mathrm{\infty })$ or $G=K_4$ and $p\in (0,\infty)$;(ii)$G=S_n$(stargraph)and$1\ge p\ge \frac{1}{2}$;$p\in (0,\frac{1}{2})$and$n\ge C(p)$or$G=S_3$and$p\in (1,\infty).$ We also find the value of the norm $\mathrm{\parallel }{M}_G{\mathrm{\parallel }}_2$ when: (i) $G=K_n$ and $n\ge 3$; (ii) $G=S_n$ and $n\ge 3.$