Let $p$ be an odd prime. From a simple undirected graph $G$, through the classical procedures of Baer (1938), Tutte (1947) and Lovász (1989), there is a $p$-group $P_G$ of class 2 and exponent $p$ that is naturally associated with $G$. Our first result is to show that this construction of groups from graphs respects isomorphism types. That is, given two graphs $G$ and $H$, $G$ and $H$ are isomorphic as graphs if and only if $P_G$ and $P_H$ are isomorphic as groups. Our second contribution is a new homomorphism notion for graphs. Based on this notion, a category of graphs can be defined, and the Baer–Lovász–Tutte construction naturally leads to a functor from this category of graphs to the category of groups.

让 $磷$成为奇素数。从一个简单的无向图 $G$，通过 Baer (1938)、Tutte (1947) 和 Lovász (1989) 的经典程序，有一个 $磷$-团体 ${磷}_G$ 类 2 和指数 $磷$ 自然而然地与 $G$. 我们的第一个结果是表明这种从图中构造的群尊重同构类型。也就是说，给定两个图 $G$ 和 $H$, $G$ 和 $H$ 同构为图当且仅当 ${磷}_G$ 和 ${磷}_H$作为群同构。我们的第二个贡献是一个新的图同态概念。基于这个概念，可以定义一类图，而 Baer-Lovász-Tutte 构造自然会导致一个函子从这类图到群的范畴。