Note: there was a formatting problem with the previous post, which I corrected. In the Markdown language, which is used here, in order to force a line break, you have to put at least two invisible trailing spaces on the line. Hey, I didn't design that feature.
Let's look at your second graph, which is simpler. Observe that because there are morphisms \\(a \rightarrow 1\\) and \\(1 \rightarrow b\\), then by composition there is also a morphism \\(a \rightarrow b\\), which is not shown in your graph. By the same reasoning, there will be edges from all of the objects to all of the objects.
These implied edges also completely change the determination of what the terminal objects are, and what the relationships of isomorphism are.
Here is the general takeaway that I see from these observations.
Every category has a graph, but not every graph comes from a category.
So before using a graph to show a point about a category, one has to first be clear about what category you intend to represent by the graph, and to check that this category is indeed represented by the graph.