Isomorphisms of graph groups

Abstract:

Given a graph $X$, define the presentation $PX$ to have generators the vertices of $X$, and a relation $xy = yx$ for each pair $x,y$ of adjacent vertices. Let $GX$ be the group with presentation $PX$, and given a field $K$, let $KX$ denote the $K$-algebra with presentation $PX$. Given graphs $X$ and $Y$ and a field $K$, it is known that the algebras $KX$ and $KY$ are isomorphic if and only if the graphs $X$ and $Y$ are isomorphic. In this paper, we use this fact to prove that if the groups $GX$ and $GY$ are isomorphic, then so are the graphs $X$ and $Y$.