The length of any longest cycle in an arbitrary graph $G$ is called the circumfrence of $G$.
While the length of any shortest cycle in an arbitrary graph $G$ is called the girth of $G$.
With that said is there a special name/notation for the length of any longest path in a graph $G$?
Also while on the topic are there any particularly special instances when the diameter of a graph is equal to the length of its longest path? For example when $G$ is a tree, I know the diameter is equal to the longest path. Though obviously the two in general are different from one and another. What about other special cases? Are there any other interesting types of graphs where the diameter is equal to the length of the longest path?
No there isn’t a special term for a longest path in a graph. You can refer to it as a longest path or a path of maximum length or a maximal path etc...