In classical homotopy theory, two spaces are considered homotopy equivalent if one space can be continuously deformed into the other. This theory, however, does not respect the discrete nature of graphs. For this reason, a discrete homotopy theory that recognizes the difference between the vertices and edges of a graph was invented. This theory is called A-homotopy theory in honor of Ron Aktin, who created the foundations of this theory in Q-analysis in the 1970s. A-homotopy theory allows us to compare graphs and to compute invariants for graphs. The intended use of these invariants is to find areas of low connectivity in a large network where information might be missing or where the network might be made more efficient.
In algebraic topology, each space $X$ has associated spaces $\widetilde{X}$ and continous maps $p: \widetilde{X} \to X$, and each pair together $(\widetilde{X}, p)$ is called a covering space of $X$. Under certain conditions, a space has a covering space with special properties, called a universal cover. Among other things, universal covers allow us to factor maps between topological spaces, which can be quite useful. In this talk, I will give a brief introduction to A-homotopy theory and discuss the universal covers I developed for graphs with no 3 or 4-cycles as well as the covering graphs obtained from quotienting these universal covers. I will end by mentioning some of the useful properties of these universal covers.