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Let $p$ be an odd prime. From a simple undirected graph $G$, through theclassical procedures of Baer (Trans. Am. Math. Soc., 1938), Tutte (J. Lond.Math. Soc., 1947) and Lov\'asz (B. Braz. Math. Soc., 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 graphsrespects 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 asgroups. Our second contribution is a new homomorphism notion for graphs. Basedon this notion, a category of graphs can be defined, and theBaer-Lov\'asz-Tutte construction naturally leads to a functor from thiscategory of graphs to the category of groups.

On the Baer-Lov\'asz-Tutte construction of groups from graphs: isomorphism types and homomorphism notions