Let me start by clearing up some confusion regarding all these names.
yWorks is the name of the company that develops and offers the yEd graph editor and several other diagramming related products.
yEd is a general purpose graph editor. This is a free-of-charge end user application.
yFiles is a programming library that offers components and features for working with graphs. This is a commercial product and is meant for software developers who need to include diagramming features in their own applications. yEd is built upon yFiles as well.
Graphity is a commercial graph editor plug-in for Atlassian Confluence (a commercial web-based wiki software). Graphity is built upon yFiles, too.
That being said, the yFiles library does offer graph analysis algorithms like shortest paths, minimum spanning tree, and several centrality measures. (It does not include algorithms for calculating eulerian cycles or chromatic numbers.) The documentation you found is for software developers that are working with the yFiles diagramming.
Unfortunately, none of yWorks' end user applications currently offer a user interface for the aforementioned analysis algorithms.
In short, yWorks does not offer an end user application with the features you are looking for.