Your attachment doesn't contain a graph object. However, perhaps it will help to mention that if you have the vertices of a polygon, you can use it to create a polyshape object.
With the polygon represented in polyshape form,
>> p=polyshape([0,0; 0 1;1 2; 1 0])
polyshape with properties:
Vertices: [4×2 double]
you can query all the kinds of things that you mentioned. For instance,