Let $A = \{1, 2, 3\}$ and consider the relation
$R = \{ (1,1), (1,2), (2,2), (3,3) \}$
Is this relation transitive?
Since $(1,2)$ and $(2,2)$ satisfy the condition $(a,b)$ and $(b,c)$ and we have $(a,c)$ which is $(1,2)$, but when you draw the graph there is no triangle loop. So $b$ and $c$ can't be equal?
Yes it is transitive. A graph is transitive if the distance between nodes is at most one, which your graph definitely is.
These 'triangles' you talk about often occur for transitive relationships, but a relationship does not have to have any triangles to be transitive. For example, the empty relation (that is, a relation whose graph has no lines at all) is perfectly transitive.
Formally, for transitivity you need that if both $(a,b) \in R$, and $(b,c) \in R$, then $(a,c) \in R$ ... This leaves it open whether or not any of $a$, $b$, or $c$ are the same element or not.
In your case, the only time we have both $(a,b)$ and $(b,c)$ is when you have $(1,1)$ and $(1,2)$ (and since you have $(1,2)$, you're good), and when you have $(1,2)$ and $(2,2)$ (and since you have $(1,2)$, you're good here as well). So, your relation is indeed transitive.