Given a finite group G having a set of generators S, the authors of this paper define a finite graph &PHgr;(G,S), both mathematically and algorithmically. Properties of these graphs and their relation to Cayley graphs are then studied.

The construction is new and interesting, and may very well be used to replace, or supplement, the more widely known Cayley graph construction in specific applications. Therefore, it deserves further attention and study. It would be nice to know more about the functorial properties of &PHgr;. Also, while the authors briefly discuss the problem of identifying the image of &PHgr; among all finite graphs, this question certainly requires further investigation.

The paper does not include specific problems in computer science that applications of this construction will solve, but this door is left wide open for those who make use of graph representations of groups. It would be good to see more research in this area.

There are a few oddities. Proposition 4.3 states that &PHgr;(G,S) is connected if and only if S is a generator set of G. Since &PHgr;(G,S) is defined only when S is a generator set, the authors seem to be saying that such graphs are always connected. Also, they use the phrase “canonic graph,” when it would be better to say “canonical graph.”