We consider infinite graphs and the associated energy forms. We show that a graph is canonically compactifiable (i.e. all functions of finite energy are bounded) if and only if the underlying set is totally bounded with respect to any finite measure intrinsic metric. Furthermore, we show that a graph is canonically compactifiable if and only if the space of functions of finite energy is an algebra. These results answer questions in a recent work of Georgakopoulos, Haeseler, Keller, Lenz, and Wojciechowski.

中文翻译：

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通过内在度量和代数性质表征可压缩图

我们考虑无限图和相关的能量形式。我们表明，当且仅当基础集相对于任何有限度量内在度量完全受限制时，图才是规范可压缩的（即，有限能量的所有函数均是有界的）。此外，我们表明，当且仅当有限能量函数的空间是代数时，图才是规范可压缩的。这些结果回答了Georgakopoulos，Haeseler，Keller，Lenz和Wojciechowski最近的工作中的问题。