Disclaimer: have not studied topology yet, currently using Wikipedia to scrap by
It just seems to me that there should be some way to connect paths in graph theory to paths (I am interested only in undirected simple graphs if it ever matters) in topology. Fix a path in some graph. Naively trying to parameterize the vertices that form the path would in general, from what I learned, result in a noncontinuous function, which is not a parameterization (thus not a topological path) by definition. Is there anyway to connect these two concepts?