Let G be a connected and finite graph with the set of vertices V and the set of edges E. Let \(M_{G}\) be the Hardy–Littlewood maximal function defined on graph G and \(M_{\upalpha ,G}\)\((0\le \upalpha <1)\) be its fractional version. In this paper, the regularity problems related to \(M_{G}\) and \(M_{\upalpha ,G}\) will be studied. We show that \(M_{G}:\mathrm{BV}_p (G)\rightarrow \mathrm{BV}_p(G)\) is bounded and \(M_{\upalpha ,G}: \ell ^p(V)\rightarrow \mathrm{BV}_q(G)\) is bounded and continuous for all \(0<p,\,q\le \infty \). Here \(\mathrm{BV}_p(G)\) is the set of all functions of bounded p-variation on V. The operator norms of \(M_{G}\) and \(M_{\upalpha ,G}\) have also been investigated.

令G为具有一组顶点V和一组边缘E的连通有限图。令\（M_ {G} \）为在图G上定义的Hardy–Littlewood最大值函数，\（M _ {\ upalpha，G} \）\（（0 \ le \ upalpha <1）\）为其分数形式。在本文中，将研究与\（M_ {G} \）和\（M _ {\ upalpha，G} \）相关的正则性问题。我们证明\（M_ {G}：\ mathrm {BV} _p（G）\ rightarrow \ mathrm {BV} _p（G）\）是有界的，而\（M _ {\ upalpha，G}：\ ell ^ p（ V）\ rightarrow \ mathrm {BV} _q（G）\）对于所有\（0 <p，\，q \ le \ infty \）是有界的和连续的。这里\（\ mathrm {BV} _p（G）\）是V上有界p变量的所有函数的集合。\（M_ {G} \）和\（M _ {\ upalpha，G} \）的算子范数也已研究。